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Journal articles on the topic 'Local and global minimizers'

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Published: 14 March 2023

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1

Giner, E. "Local minimizers of integral functionals are global minimizers." Proceedings of the American Mathematical Society 123, no. 3 (March 1, 1995): 755. http://dx.doi.org/10.1090/s0002-9939-1995-1254839-1.

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2

Jimbo, Shuichi, and Jian Zhai. "Domain perturbation method and local minimizers to Ginzburg-Landau functional with magnetic effect." Abstract and Applied Analysis 5, no. 2 (2000): 101–12. http://dx.doi.org/10.1155/s1085337500000233.

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We prove the existence of vortex local minimizers to Ginzburg-Landau functional with a global magnetic effect. A domain perturbating method is developed, which allows us to extend a local minimizer on a nonsimply connected superconducting material to the local minimizer with vortex on a simply connected material.

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3

Andersson, Mats, Oleg Burdakov, Hans Knutsson, and Spartak Zikrin. "Global Search Strategies for Solving Multilinear Least-Squares Problems." Sultan Qaboos University Journal for Science [SQUJS] 16 (April 1, 2012): 12. http://dx.doi.org/10.24200/squjs.vol17iss1pp12-21.

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The multilinear least-squares (MLLS) problem is an extension of the linear least-squares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows for moving from one local minimizer to a better one. The efficiency of this strategy is illustrated by the results of numerical experiments performed for some problems related to the design of filter networks.

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4

Rodríguez, Nancy, and Yi Hu. "On the steady-states of a two-species non-local cross-diffusion model." Journal of Applied Analysis 26, no. 1 (June 1, 2020): 1–19. http://dx.doi.org/10.1515/jaa-2020-2003.

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AbstractWe investigate the existence and properties of steady-state solutions to a degenerate, non-local system of partial differential equations that describe two-species segregation in homogeneous and heterogeneous environments. This is accomplished via the analysis of the existence and non-existence of global minimizers to the corresponding free energy functional. We prove that in the spatially homogeneous case global minimizers exist if and only if the mass of the potential governing the intra-species attraction is sufficiently large and the support of the potential governing the interspecies repulsion is bounded. Moreover, when they exist they are such that the two species have disjoint support, leading to complete segregation. For the heterogeneous environment we show that if a sub-additivity condition is satisfied then global minimizers exists. We provide an example of an environment that leads to the sub-additivity condition being satisfied. Finally, we explore the bounded domain case with periodic conditions through the use of numerical simulations.

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5

Palatucci, Giampiero, Ovidiu Savin, and Enrico Valdinoci. "Local and global minimizers for a variational energy involving a fractional norm." Annali di Matematica Pura ed Applicata 192, no. 4 (January 4, 2012): 673–718. http://dx.doi.org/10.1007/s10231-011-0243-9.

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6

Porretta, Alessio. "On the regularity of the total variation minimizers." Communications in Contemporary Mathematics 23, no. 01 (November 20, 2019): 1950082. http://dx.doi.org/10.1142/s0219199719500822.

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We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by Rudin, Osher and Fatemi. In particular, we show that if the source term [Formula: see text] is locally (respectively, globally) Lipschitz, then the solution has the same regularity with local (respectively, global) Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by Caselles, Chambolle and Novaga for dimension [Formula: see text] and (in case of the global regularity) for convex domains.

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7

Teughels, Anne, Guido De Roeck, and Johan A. K. Suykens. "Global optimization by coupled local minimizers and its application to FE model updating." Computers & Structures 81, no. 24-25 (September 2003): 2337–51. http://dx.doi.org/10.1016/s0045-7949(03)00313-4.

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8

AZORERO, J. P. GARCÍA, I. PERAL ALONSO, and JUAN J. MANFREDI. "SOBOLEV VERSUS HÖLDER LOCAL MINIMIZERS AND GLOBAL MULTIPLICITY FOR SOME QUASILINEAR ELLIPTIC EQUATIONS." Communications in Contemporary Mathematics 02, no. 03 (August 2000): 385–404. http://dx.doi.org/10.1142/s0219199700000190.

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9

Enkhbat, R., and T. Bayartugs. "Quasiconvex Semidefinite Minimization Problem." Journal of Optimization 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/346131.

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We introduce so-called semidefinite quasiconvex minimization problem. We derive new global optimality conditions for the above problem. Based on the global optimality conditions, we construct an algorithm which generates a sequence of local minimizers which converge to a global solution.

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10

Duboscq, Romain, and Olivier Pinaud. "On local quantum Gibbs states." Journal of Mathematical Physics 63, no. 10 (October 1, 2022): 102102. http://dx.doi.org/10.1063/5.0058574.

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We address in this work the problem of minimizing quantum entropies under local constraints. We suppose that macroscopic quantities, such as the particle density, current, and kinetic energy, are fixed at each point of [Formula: see text] and look for a density operator over [Formula: see text], minimizing an entropy functional. Such minimizers are referred to as local Gibbs states. This setting is in contrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises, for instance, in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.

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11

Taati, Akram, and Maziar Salahi. "On Local Non-Global Minimizers of Quadratic Optimization Problem with a Single Quadratic Constraint." Numerical Functional Analysis and Optimization 41, no. 8 (March 10, 2020): 969–1005. http://dx.doi.org/10.1080/01630563.2020.1733605.

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12

DÁVILA, JUAN. "GLOBAL REGULARITY FOR A SINGULAR EQUATION AND LOCAL H1 MINIMIZERS OF A NONDIFFERENTIABLE FUNCTIONAL." Communications in Contemporary Mathematics 06, no. 01 (February 2004): 165–93. http://dx.doi.org/10.1142/s0219199704001240.

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We prove optimal Hölder estimates up to the boundary for the maximal solution of a singular elliptic equation. The techniques used in this argument are applied to show that in some situations the maximal solution is a local minimizer of the corresponding functional in the topology of H1.

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13

Li, Xinrong, Naihua Xiu, and Shenglong Zhou. "Matrix Optimization Over Low-Rank Spectral Sets: Stationary Points and Local and Global Minimizers." Journal of Optimization Theory and Applications 184, no. 3 (December 9, 2019): 895–930. http://dx.doi.org/10.1007/s10957-019-01606-8.

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14

Li, Mei Xia. "A Class of Augumented Lagrangian Function for Nonlinear Programming." Advanced Materials Research 271-273 (July 2011): 1955–60. http://dx.doi.org/10.4028/www.scientific.net/amr.271-273.1955.

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In this paper, we discuss an exact augumented Lagrangian functions for the non- linear programming problem with both equality and inequality constraints, which is the gen- eration of the augmented Lagrangian function in corresponding reference only for inequality constraints nonlinear programming problem. Under suitable hypotheses, we give the relation- ship between the local and global unconstrained minimizers of the augumented Lagrangian function and the local and global minimizers of the original constrained problem. From the theoretical point of view, the optimality solution of the nonlinear programming with both equality and inequality constraints and the values of the corresponding Lagrangian multipli- ers can be found by the well known method of multipliers which resort to the unconstrained minimization of the augumented Lagrangian function presented in this paper.

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15

Chen, Yifan, Yuejiao Sun, and Wotao Yin. "Run-and-Inspect Method for nonconvex optimization and global optimality bounds for R-local minimizers." Mathematical Programming 176, no. 1-2 (April 27, 2019): 39–67. http://dx.doi.org/10.1007/s10107-019-01397-w.

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16

Bakir, Pelin Gundes, Edwin Reynders, and Guido De Roeck. "An improved finite element model updating method by the global optimization technique ‘Coupled Local Minimizers’." Computers & Structures 86, no. 11-12 (June 2008): 1339–52. http://dx.doi.org/10.1016/j.compstruc.2007.08.009.

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17

Pál, László, Tibor Csendes, Mihály Csaba Markót, and Arnold Neumaier. "Black Box Optimization Benchmarking of the GLOBAL Method." Evolutionary Computation 20, no. 4 (December 2012): 609–39. http://dx.doi.org/10.1162/evco_a_00089.

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GLOBAL is a multi-start type stochastic method for bound constrained global optimization problems. Its goal is to find the best local minima that are potentially global. For this reason it involves a combination of sampling, clustering, and local search. The role of clustering is to reduce the number of local searches by forming groups of points around the local minimizers from a uniformly sampled domain and to start few local searches in each of those groups. We evaluate the performance of the GLOBAL algorithm on the BBOB 2009 noiseless testbed, containing problems which reflect the typical difficulties arising in real-world applications. The obtained results are also compared with those obtained form the simple multi-start procedure in order to analyze the effects of the applied clustering rule. An improved parameterization is introduced in the GLOBAL method and the performance of the new procedure is compared with the performance of the MATLAB GlobalSearch solver by using the BBOB 2010 test environment.

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18

do Nascimento, Arnaldo Simal, and João Biesdorf. "Global and Local Minimizers of the Cahn-Hilliard Functional Over a Parallelepiped: With And Without Constraint." Milan Journal of Mathematics 79, no. 1 (June 2011): 303–10. http://dx.doi.org/10.1007/s00032-011-0157-4.

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19

Bi, Yingjie, Haixiang Zhang, and Javad Lavaei. "Local and Global Linear Convergence of General Low-Rank Matrix Recovery Problems." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 9 (June 28, 2022): 10129–37. http://dx.doi.org/10.1609/aaai.v36i9.21252.

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We study the convergence rate of gradient-based local search methods for solving low-rank matrix recovery problems with general objectives in both symmetric and asymmetric cases, under the assumption of the restricted isometry property. First, we develop a new technique to verify the Polyak-Lojasiewicz inequality in a neighborhood of the global minimizers, which leads to a local linear convergence region for the gradient descent method. Second, based on the local convergence result and a sharp strict saddle property proven in this paper, we present two new conditions that guarantee the global linear convergence of the perturbed gradient descent method. The developed local and global convergence results provide much stronger theoretical guarantees than the existing results. As a by-product, this work significantly improves the existing bounds on the RIP constant required to guarantee the non-existence of spurious solutions.

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20

Ghisi, Marina, Massimo Gobbino, and Giulio Rovellini. "Symmetry-breaking in a generalized Wirtinger inequality." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 4 (October 2018): 1381–94. http://dx.doi.org/10.1051/cocv/2017059.

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The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the p-norm of the derivative among all functions whose q-norm is equal to 1 and whose (r − 1)-power has zero average. Symmetry properties of minimizers have attracted great attention in mathematical literature in the last decades, leading to a precise characterization of symmetry and asymmetry regions. In this paper we provide a proof of the symmetry result without computer assisted steps, and a proof of the asymmetry result which works as well for local minimizers. As a consequence, we have now a full elementary description of symmetry and asymmetry cases, both for global and for local minima. Proofs rely on appropriate nonlinear variable changes.

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21

Kerk, Lee Chang, and Rohanin Ahmad. "Algorithm for Solution of Non-convex Optimization Problem Through Piece-wise Convex Transformation." MATEMATIKA 34, no. 2 (December 2, 2018): 381–92. http://dx.doi.org/10.11113/matematika.v34.n2.977.

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Optimization is central to any problem involving decision making. Thearea of optimization has received enormous attention for over 30 years and it is still popular in research field to this day. In this paper, a global optimization method called Kerk and Rohanin’s Trusted Interval will be introduced. The method introduced is able to identify all local solutions by converting non-convex optimization problems into piece-wise convex optimization problems. A mechanism which only considers the convex part where minimizers existed on a function is applied. This mechanism allows the method to filter out concave parts and some unrelated parts automatically. The identified convex parts are called trusted intervals. The descent property and the globally convergent of the method was shown in this paper. 15 test problems have been used to show the ability of the algorithm proposed in locating global minimizer.

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22

Külske, Christof, and Daniel Meißner. "Stable and Metastable Phases for the Curie–Weiss–Potts Model in Vector-Valued Fields via Singularity Theory." Journal of Statistical Physics 181, no. 3 (August 1, 2020): 968–89. http://dx.doi.org/10.1007/s10955-020-02615-y.

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Abstract We study the metastable minima of the Curie–Weiss Potts model with three states, as a function of the inverse temperature, and for arbitrary vector-valued external fields. Extending the classic work of Ellis and Wang (Stoch Process Appl 35(1):59–79, 1990) and Wang (Stoch Process Appl 50(2):245–252, 1994) we use singularity theory to provide the global structure of metastable (or local) minima. In particular, we show that the free energy has up to four local minimizers (some of which may at the same time be global) and describe the bifurcation geometry of their transitions under variation of the parameters.

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23

Sawada, Osamu. "Scalarity and alternatives of Japanese mora (letter)-based minimizers." Proceedings of the Linguistic Society of America 4, no. 1 (March 15, 2019): 21. http://dx.doi.org/10.3765/plsa.v4i1.4525.

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This paper investigates interpretations of the Japanese mora- (letter-) based minimizer “X.Y.Z”-no “X”-no ji-mo ‘even the letter “X” of “X.Y.Z”.’ I argue that this mora/letter-based minimizer has two types, a literal type and a non-literal type, and each type has different semantic characteristics regarding scale structure and computation of alternatives. In the literal type, X corresponds to the first mora of a target “X.Y.Z” and is construed as a minimum on the number scale of moras (among higher scalar alternatives). On the other hand, in the non-literal type it refers to the degree of a main predicate about the target “X.Y.Z” where X is construed as a minimum on the scale of the main predicate. That is, in the non-literal type, scale does not have to do with the number of moras, but with the degree of a predicate. I propose on the basis of the findings that in addition to a local minimizer whose alternatives are lexically activated (Chierchia 2013), there is a global minimizer in natural language, whose alternatives are activated by information contained in the main predicate.

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24

Segatti, Antonio, Michael Snarski, and Marco Veneroni. "Analysis of a variational model for nematic shells." Mathematical Models and Methods in Applied Sciences 26, no. 10 (August 25, 2016): 1865–918. http://dx.doi.org/10.1142/s0218202516500470.

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We analyze an elastic surface energy which was recently introduced by G. Napoli and L. Vergori to model thin films of nematic liquid crystals. We show how a novel approach in modeling the surface’s extrinsic geometry leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: (i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; (ii) we prove, by a Ginzburg–Landau approximation, the well-posedness of the gradient flow of the energy; (iii) in the case of a parametrized torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.

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25

Raus, Toomas, and Uno Hämarik. "Q-Curve and Area Rules for Choosing Heuristic Parameter in Tikhonov Regularization." Mathematics 8, no. 7 (July 16, 2020): 1166. http://dx.doi.org/10.3390/math8071166.

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We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the global minimizer of the quasi-optimality function. In some problems this rule fails. We prove that one of the local minimizers of the quasi-optimality function is always a good regularization parameter. For the choice of the proper local minimizer we propose to construct the Q-curve which is the analogue of the L-curve, but on the x-axis we use modified discrepancy instead of discrepancy and on the y-axis the quasi-optimality function instead of the norm of the approximate solution. In the area rule we choose for the regularization parameter such local minimizer of the quasi-optimality function for which the area of the polygon, connecting on Q-curve this minimum point with certain maximum points, is maximal. We also provide a posteriori error estimates of the approximate solution, which allows to check the reliability of the parameter chosen heuristically. Numerical experiments on an extensive set of test problems confirm that the proposed rules give much better results than previous heuristic rules. Results of proposed rules are comparable with results of the discrepancy principle and the monotone error rule, if the last two rules use the exact noise level.

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26

Farshbaf-Shaker, M. Hassan, and Christian Heinemann. "A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media." Mathematical Models and Methods in Applied Sciences 25, no. 14 (October 14, 2015): 2749–93. http://dx.doi.org/10.1142/s0218202515500608.

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In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with non-homogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.

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27

Feng, Xue, Chunlin Wu, and Chao Zeng. "On the local and global minimizers of $ \newcommand{\e}{{\rm e}} \ell_0$ gradient regularized model with box constraints for image restoration." Inverse Problems 34, no. 9 (July 23, 2018): 095007. http://dx.doi.org/10.1088/1361-6420/aad1c5.

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28

Wang, Wei-xiang, You-lin Shang, and Ying Zhang. "Global Minimization of Nonsmooth Constrained Global Optimization with Filled Function." Mathematical Problems in Engineering 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/563860.

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A novel filled function is constructed to locate a global optimizer or an approximate global optimizer of smooth or nonsmooth constrained global minimization problems. The constructed filled function contains only one parameter which can be easily adjusted during the minimization. The theoretical properties of the filled function are discussed and a corresponding solution algorithm is proposed. The solution algorithm comprises two phases: local minimization and filling. The first phase minimizes the original problem and obtains one of its local optimizers, while the second phase minimizes the constructed filled function and identifies a better initial point for the first phase. Some preliminary numerical results are also reported.

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29

Arutyunov, Aram V., Dmitry Yu Karamzin, and Fernando Lobo Pereira. "Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints." Axioms 11, no. 2 (January 20, 2022): 40. http://dx.doi.org/10.3390/axioms11020040.

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This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based on the Robinson stability theorem is derived. The main novelty of this approach compared to the known results in this area is that only a local regularity with respect to the mixed constraints, that is, a regularity in an ε-tube about the minimizer, is required instead of the conventional stronger global regularity hypothesis. This affects the maximum condition. Therefore, the normal set of Lagrange multipliers in question satisfies the maximum principle, albeit along with the modified maximum condition, in which the maximum is taken over a reduced feasible set. In the second part of this work, we address the case of abnormal minimizers, that is, when the full rank of controllability matrix condition is not valid. The same type of reduced maximum condition is obtained.

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30

Bester, Nancy L. "Incorporating Energy Criteria in Intermodal Transportation Policy Decisions." Transportation Research Record: Journal of the Transportation Research Board 1522, no. 1 (January 1996): 83–86. http://dx.doi.org/10.1177/0361198196152200111.

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Regional and local governments are collectively responsible for maintaining the economic health of their communities and managing traffic congestion, air quality, land use, and other related growth-management issues. Yet global climate change and air quality problems result from the consumption of energy in the production of goods and services that help sustain the economy. Public policy solutions to such problems are often difficult to design because of the interrelated nature of the environment, economic activities, and the infrastructure that links them together. A conceptual framework for thinking about the market behavior of consumers and producers as cost minimizers and offering a new way to design public policies using economic and energy efficiency goals is presented for the use of public-policy makers. Production theory can be used to explain how land, vehicles, infrastructure, and energy are combined to produce transportation goods and services. Heat and waste by-products from the production process act as the precursors of air pollution and other global climate-change problems. If public policies are designed to minimize such problems, policy analysis methods need to include those factors that help determine the cost and benefits of prospective policy alternatives, as well as information on how the net benefits of such policies are redistributed in society. A list of criteria to use in selecting analysis methods for this purpose is suggested.

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31

Alnowibet, Khalid Abdulaziz, Ahmad M. Alshamrani, Adel Fahad Alrasheedi, Salem Mahdi, Mahmoud El-Alem, Abdallah Aboutahoun, and Ali Wagdy Mohamed. "Efficient Modified Meta-Heuristic Technique for Unconstrained Optimization Problems." Axioms 11, no. 9 (September 19, 2022): 483. http://dx.doi.org/10.3390/axioms11090483.

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In this paper, a new Modified Meta-Heuristic algorithm is proposed. This method contains some modifications to improve the performance of the simulated-annealing algorithm (SA). Most authors who deal with improving the SA algorithm presented some improvements and modifications to one or more of the five standard features of the SA algorithm. In this paper, we improve the SA algorithm by presenting some suggestions and modifications to all five standard features of the SA algorithm. Through these suggestions and modifications, we obtained a new algorithm that finds the approximate solution to the global minimum of a non-convex function. The new algorithm contains novel parameters, which are updated at each iteration. Therefore, the variety and alternatives in choosing these parameters demonstrated a noticeable impact on the performance of the proposed algorithm. Furthermore, it has multiple formulas by which the candidate solutions are generated. Diversity in these formulas helped the proposed algorithm to escape a local point while finding the global minimizer of a non-convex function. The efficiency of the proposed algorithm is reported through extensive numerical experiments on some well-known test problems. The performance profiles are used to evaluate and compare the performance of our proposed algorithm against the other five meta-heuristic algorithms. The comparison results between the performance of our suggested algorithm and the other five algorithms indicate that the proposed algorithm is competitive with, and in all cases superior to, the five algorithms in terms of the efficiency, reliability, and effectiveness for finding the global minimizers of non-convex functions. This superiority of the new proposed algorithm is due to those five modified standard features.

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32

Kok, Schalk, and Carl Sandrock. "Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function." Evolutionary Computation 17, no. 3 (September 2009): 437–53. http://dx.doi.org/10.1162/evco.2009.17.3.437.

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Two variants of the extended Rosenbrock function are analyzed in order to find the stationary points. The first variant is shown to possess a single stationary point, the global minimum. The second variant has numerous stationary points for high dimensionality. A previously proposed method is shown to be numerically intractable, requiring arbitrary precision computation in many cases to enumerate candidate solutions. Instead, a standard Newtonian method with multi-start is applied to locate stationary points. The relative magnitude of the negative and positive eigenvalues of the Hessian is also computed, in order to characterize the saddle points. For dimensions up to 100, only two local minimizers are found, but many saddle points exist. Two saddle points with a single negative eigenvalue exist for high dimensionality, which may appear as “near” local minima. The remaining saddle points we found have a predictable form, and a method is proposed to estimate their number. Monte Carlo simulation indicates that it is unlikely to escape these saddle points using uniform random search. A standard particle swarm algorithm also struggles to improve upon a saddle point contained within the initial population.

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33

Li, Guoquan, and Yan Wang. "Global Optimality Conditions for Nonlinear Programming Problems with Linear Equality Constraints." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/213178.

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Some necessary global optimality conditions and sufficient global optimality conditions for nonconvex minimization problems with a quadratic inequality constraint and a linear equality constraint are derived. In particular, global optimality conditions for nonconvex minimization over a quadratic inequality constraint which extend some known global optimality conditions in the existing literature are presented. Some numerical examples are also given to illustrate that a global minimizer satisfies the necessary global optimality conditions but a local minimizer which is not global may fail to satisfy them.

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34

Barabasz, Barbara, Ewa Gajda-Zagórska, Stanisław Migórski, Maciej Paszyński, Robert Schaefer, and Maciej Smołka. "A hybrid algorithm for solving inverse problems in elasticity." International Journal of Applied Mathematics and Computer Science 24, no. 4 (December 1, 2014): 865–86. http://dx.doi.org/10.2478/amcs-2014-0064.

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Abstract The paper offers a new approach to handling difficult parametric inverse problems in elasticity and thermo-elasticity, formulated as global optimization ones. The proposed strategy is composed of two phases. In the first, global phase, the stochastic hp-HGS algorithm recognizes the basins of attraction of various objective minima. In the second phase, the local objective minimizers are closer approached by steepest descent processes executed singly in each basin of attraction. The proposed complex strategy is especially dedicated to ill-posed problems with multimodal objective functionals. The strategy offers comparatively low computational and memory costs resulting from a double-adaptive technique in both forward and inverse problem domains. We provide a result on the Lipschitz continuity of the objective functional composed of the elastic energy and the boundary displacement misfits with respect to the unknown constitutive parameters. It allows common scaling of the accuracy of solving forward and inverse problems, which is the core of the introduced double-adaptive technique. The capability of the proposed method of finding multiple solutions is illustrated by a computational example which consists in restoring all feasible Young modulus distributions minimizing an objective functional in a 3D domain of a photo polymer template obtained during step and flash imprint lithography.

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35

Lin, Hongwei, Yuping Wang, and Xiaoli Wang. "An Auxiliary Function Method for Global Minimization in Integer Programming." Mathematical Problems in Engineering 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/402437.

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An auxiliary function method is proposed for finding the global minimizer of integer programming problem. Firstly, we propose a method to transform the original problem into an integer programming with box constraint, which does not change the properties of the original problem. For the transformed problem, we propose an auxiliary function to escape from the current local minimizer and to get a better one. Then, based on the proposed auxiliary function, a new algorithm to find the global minimizer of integer programming is proposed. At last, numerical results are given to demonstrate the effectiveness and efficiency of the proposed method.

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36

Wu, Boying, and Yunyun Yang. "Local- and Global-Statistics-Based Active Contour Model for Image Segmentation." Mathematical Problems in Engineering 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/791958.

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This paper presents a local- and global-statistics-based active contour model for image segmentation by applying the globally convex segmentation method. We first propose a convex energy functional with a local-Gaussian-distribution-fitting term with spatially varying means and variances and an auxiliary global-intensity-fitting term. A weight function that varies dynamically with the location of the image is applied to adjust the weight of the global-intensity-fitting term dynamically. The weighted total variation norm is incorporated into the energy functional to detect boundaries easily. The split Bregman method is then applied to minimize the proposed energy functional more efficiently. Our model has been applied to synthetic and real images with promising results. With the local-Gaussian-distribution-fitting term, our model can also handle some texture images. Comparisons with other models show the advantages of our model.

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37

Khan, Akhtar A., and Dumitru Motreanu. "Local minimizers versus X-local minimizers." Optimization Letters 7, no. 5 (April 7, 2012): 1027–33. http://dx.doi.org/10.1007/s11590-012-0474-8.

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Wang, Wei-Xiang, You-Lin Shang, and Lian-Sheng Zhang. "Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming." Discrete Dynamics in Nature and Society 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/171697.

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This paper presents a filled function method for finding a global optimizer of integer programming problem. The method contains two phases: the local minimization phase and the filling phase. The goal of the former phase is to identify a local minimizer of the objective function, while the filling phase aims to search for a better initial point for the first phase with the aid of the filled function. A two-parameter filled function is proposed, and its properties are investigated. A corresponding filled function algorithm is established. Numerical experiments on several test problems are performed, and preliminary computational results are reported.

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ZHAO, YU QIAN, XIAO FANG WANG, FRANK Y. SHIH, and GANG YU. "A LEVEL-SET METHOD BASED ON GLOBAL AND LOCAL REGIONS FOR IMAGE SEGMENTATION." International Journal of Pattern Recognition and Artificial Intelligence 26, no. 01 (February 2012): 1255004. http://dx.doi.org/10.1142/s021800141255004x.

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This paper presents a new level-set method based on global and local regions for image segmentation. First, the image fitting term of Chan and Vese (CV) model is adapted to detect the image's local information by convolving a Gaussian kernel function. Then, a global term is proposed to detect large gradient amplitude at the outer region. The new energy function consists of both local and global terms, and is minimized by the gradient descent method. Experimental results on both synthetic and real images show that the proposed method can detect objects in inhomogeneous, low-contrast, and noisy images more accurately than the CV model, the local binary fitting model, and the Lankton and Tannenbaum model.

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Grigoriev, L., and M. Salikhov. "Financial Crisis-2008: Entering Global Recession." Voprosy Ekonomiki, no. 12 (December 20, 2008): 27–45. http://dx.doi.org/10.32609/0042-8736-2008-12-27-45.

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Main factors and development of the global financial crisis-2008 are generally discussed in the paper. The downturn in one of the local sectors of the US economy has caused major threats to functioning global financial markets. Structural problems of the Russian financial sector ("illusion of adequacy") have greatly enhanced negative consequences of the global crisis for the Russian economy. On the global level, main steps to minimize the costs of the crisis should deal with limiting protectionism growth, coordinating measures of economic policy and preventing a hard landing of a large group of economies.

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PARK, CHEONG HEE. "IMPROVED ALGORITHMS FOR UNSUPERVISED DISCRIMINANT PROJECTION." International Journal of Pattern Recognition and Artificial Intelligence 24, no. 02 (March 2010): 193–206. http://dx.doi.org/10.1142/s0218001410007920.

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Dimension reduction has been applied in various areas of pattern recognition and data mining. While a traditional dimension reduction method, Principal Component Analysis (PCA) finds projective directions to maximize the global scatter in data, Locality Preserving Projection (LPP) pursues linear dimension reduction to minimize the local scatter. However, the discriminative power by either global or local scatter optimization is not guaranteed to be effective for classification. A recently proposed method, Unsupervised Discriminant Projection (UDP) aims to minimize the local scatter among near points and maximize the global scatter of distant points at the same time. Although its performance has been proven to be comparable to other dimension reduction methods, PCA preprocessing step due to the singularity of global and local scatter matrices may degrade the performance of UDP. In this paper, we propose several algorithms to improve the performances of UDP greatly. An improved algorithm for UDP is presented which applies the Generalized Singular Value Decomposition (GSVD) to overcome singularities of scatter matrices in UDP. Two-dimensional UDP and nonlinear extension of UDP are also proposed. Extensive experimental results demonstrate superiority of the proposed algorithms.

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Umar, Abubakar, Zhanqun Shi, Alhadi Khlil, and Zulfiqar I. B. Farouk. "Developing a New Robust Swarm-Based Algorithm for Robot Analysis." Mathematics 8, no. 2 (January 22, 2020): 158. http://dx.doi.org/10.3390/math8020158.

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Metaheuristics are incapable of analyzing robot problems without being enhanced, modified, or hybridized. Enhanced metaheuristics reported in other works of literature are problem-specific and often not suitable for analyzing other robot configurations. The parameters of standard particle swarm optimization (SPSO) were shown to be incapable of resolving robot optimization problems. A novel algorithm for robot kinematic analysis with enhanced parameters is hereby presented. The algorithm is capable of analyzing all the known robot configurations. This was achieved by studying the convergence behavior of PSO under various robot configurations, with a view of determining new PSO parameters for robot analysis and a suitable adaptive technique for parameter identification. Most of the parameters tested stagnated in the vicinity of strong local minimizers. A few parameters escaped stagnation but were incapable of finding the global minimum solution, this is undesirable because accuracy is an important criterion for robot analysis and control. The algorithm was trained to identify stagnating solutions. The algorithm proposed herein was found to compete favorably with other algorithms reported in the literature. There is a great potential of further expanding the findings herein for dynamic parameter identification.

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Goncalves, Douglas S., Marcia A. Gomes-Ruggiero, Carlile Lavor, Osvaldo J. Farias, and P. H. Souto Ribeiro. "Local solutions of maximum likelihood estimation in quantum state tomography." Quantum Information and Computation 12, no. 9&10 (September 2012): 775–90. http://dx.doi.org/10.26421/qic12.9-10-4.

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Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.

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Winkert, Patrick. "Local -minimizers versus local -minimizers of nonsmooth functionals." Nonlinear Analysis: Theory, Methods & Applications 72, no. 11 (June 2010): 4298–303. http://dx.doi.org/10.1016/j.na.2010.02.006.

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CHOI, CHANGKYU, and JU-JANG LEE. "FINDING MULTIPLE LOCAL MINIMA USING CHAOTIC JUMP." International Journal of Cooperative Information Systems 07, no. 01 (March 1998): 105–15. http://dx.doi.org/10.1142/s0218843098000064.

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In this paper, the local minima free search algorithm using chaos is proposed for an unstructured search space. The problem is that given the quality function, find the value of a configuration that minimizes the quality function. The proposed algorithm started basically from the gradient search technique but at the prescribed points, that is, local minimum points, which are to be automatically detected the chaotic jump is introduced by the dynamics of a chaotic neuron. Chaotic motions are mainly because of the Gaussian function having a hysteresis as a refractoriness. In order to enhance the probability of finding the global minimum, a parallel search strategy is also given. The validity of the proposed method wil be verified in simulation examples of the function minimization problem and the motion planning problem of a mobile robot.

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Hernández, Jaime, and Xavier Emery. "A geostatistical approach to optimize sampling designs for local forest inventories." Canadian Journal of Forest Research 39, no. 8 (August 2009): 1465–74. http://dx.doi.org/10.1139/x09-048.

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In forest management, it is of interest to obtain detailed inventories such that the local prediction errors on forest attributes are less than a prespecified threshold, while keeping the number of ground samples as low as possible. Given an initial sampling design, we propose an algorithm to determine the additional sample locations. The algorithm relies on two tools: geostatistical simulation, which allows measuring the uncertainty in the values of the attribute of interest, and simulated annealing, which allows finding an infill design that minimizes a given objective function. The proposed approach is applied to a data set from a Prosopis spp. plantation located in the Atacama Desert, in which the measured attribute is the rate of tree survival.

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Park, Byung Joo, Farkhod A. Alisherov, and Byeong Yun Chang. "Efficient Mobile Anchor Point Messaging Assumption in IPv6 Based Wireless Mobile Networks." Advanced Materials Research 267 (June 2011): 1038–43. http://dx.doi.org/10.4028/www.scientific.net/amr.267.1038.

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Using a hierarchy that differentiates local mobility from global mobility is more appropriate to the Internet because it improves handoff performance, minimizes the loss of packets that may occur during transition and significantly reduces the mobility management signaling load on the Internet. But the only disadvantage of the Hierarchical Mobile IPv6 is that Mobility Anchor Point are sometimes "far away" and there is too much signaling from the mobile node. This paper assumes a new messaging architecture for the Hierarchical Mobile IPv6, which can shorten the signaling from the mobile node and minimize the handoff latency.

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Avramenko, S. E., T. A. Zheldak, and L. S. Koriashkina. "GUIDED HYBRID GENETIC ALGORITHM FOR SOLVING GLOBAL OPTIMIZATION PROBLEMS." Radio Electronics, Computer Science, Control, no. 2 (July 10, 2021): 174–88. http://dx.doi.org/10.15588/1607-3274-2021-2-18.

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Context. One of the leading problems in the world of artificial intelligence is the optimization of complex systems, which is often represented as a nonlinear function that needs to be minimized. Such functions can be multimodal, non-differentiable, and even set as a black box. Building effective methods for solving global optimization problems raises great interest among scientists. Objective. Development of a new hybrid genetic algorithm for solving global optimization problems, which is faster than existing analogues. Methods. One of the crucial challenges for hybrid methods in solving nonlinear global optimization problems is the rational use of local search, as its application is accompanied by quite expensive computational costs. This paper proposes a new GBOHGA hybrid genetic algorithm that reproduces guided local search and combines two successful modifications of genetic algorithms. The first one is BOHGA that establishes a qualitative balance between local and global search. The second one is HGDN that prevents reexploration of the previously explored areas of a search space. In addition, a modified bump-function and an adaptive scheme for determining one of its parameters – the radius of the “deflation” of the objective function in the vicinity of the already found local minimum – were presented to accelerate the algorithm. Results. GBOHGA performance compared to other known stochastic search heuristics on a set of 33 test functions in 5 and 25dimensional spaces. The results of computational experiments indicate the competitiveness of GBOHGA, especially in problems with multimodal functions and a large number of variables. Conclusions. The new GBOHGA hybrid algorithm, developed on the basis of the integration of guided local search ideas and BOHGA and HGDN algorithms, allows to save significant computing resources and speed up the solution process of the global optimization problem. It should be used to solve global optimization problems that arise in engineering design, solving organizational and management problems, especially when the mathematical model of the problem is complex and multidimensional.

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Ma, Ziye, Yingjie Bi, Javad Lavaei, and Somayeh Sojoudi. "Sharp Restricted Isometry Property Bounds for Low-Rank Matrix Recovery Problems with Corrupted Measurements." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 7 (June 28, 2022): 7672–81. http://dx.doi.org/10.1609/aaai.v36i7.20734.

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In this paper, we study a general low-rank matrix recovery problem with linear measurements corrupted by some noise. The objective is to understand under what conditions on the restricted isometry property (RIP) of the problem local search methods can find the ground truth with a small error. By analyzing the landscape of the non-convex problem, we first propose a global guarantee on the maximum distance between an arbitrary local minimizer and the ground truth under the assumption that the RIP constant is smaller than 1/2. We show that this distance shrinks to zero as the intensity of the noise reduces. Our new guarantee is sharp in terms of the RIP constant and is much stronger than the existing results. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. Next, we prove the strict saddle property, which guarantees the global convergence of the perturbed gradient descent method in polynomial time. The developed results demonstrate how the noise intensity and the RIP constant of the problem affect the landscape of the problem.

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Brown, Dana, and Jette Steen Knudsen. "Managing corporate responsibility globally and locally: Lessons from a CR leader." Business and Politics 14, no. 3 (October 2012): 1–29. http://dx.doi.org/10.1515/bap-2012-0021.

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Corporate Responsibility (CR) is today an essential component of corporate global strategy. CR can bolster the institutional context for market expansion fill institutional voids or facilitate market entry as a component of non-market strategy. Yet, in fulfilling these functions, CR may need to be highly sensitive to local contexts. How can transnational firms organize CR so as to maximize efficiencies from globalization and to minimize the fragmentation of corporate organizational cultures? provide a framework for analyzing the way that corporations coordinate global and local functions. We build on this framework in a case study of Novo Nordisk and its approach to determining global and local CR policies and procedures with regard to its China and US subsidiaries. Our findings suggest that it is important for companies to define a common set of organizational norms. In addition, CR need to be sensitive to local institutional contexts, but learning from subsidiary experience is important and lends itself to standardization and replication of initiatives across market contexts.

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