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Journal articles on the topic 'Nonconvex Sets'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Cong, Chang, and Peibiao Zhao. "Non-Cash Risk Measure on Nonconvex Sets." Mathematics 6, no. 10 (October 1, 2018): 186. http://dx.doi.org/10.3390/math6100186.

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Monetary risk measures are interpreted as the smallest amount of external cash that must be added to a financial position to make the position acceptable. In this paper, A new concept: non-cash risk measure is proposed and this measure provides an approach to transform the unacceptable positions into the acceptable positions in a nonconvex set. Non-cash risk measure uses not only cash but also other kinds of assets to adjust the position. This risk measure is nonconvex due to the use of optimization problem in L 1 norm. A convex extension of the nonconvex risk measure is derived and the relationship between the convex extension and the non-cash risk measure is detailed.

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2

Blasi, Francesco S. de, Giulio Pianigiani, and Vasile Staicu. "On the solution sets of some nonconvex hyperbolic differential inclusions." Czechoslovak Mathematical Journal 45, no. 1 (1995): 107–16. http://dx.doi.org/10.21136/cmj.1995.128505.

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3

Panasyuk, A. I. "Differential equation for nonconvex attainment sets." Mathematical Notes of the Academy of Sciences of the USSR 37, no. 5 (May 1985): 395–400. http://dx.doi.org/10.1007/bf01157972.

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4

Chrétien, Stéphane, and Pascal Bondon. "Projection Methods for Uniformly Convex Expandable Sets." Mathematics 8, no. 7 (July 6, 2020): 1108. http://dx.doi.org/10.3390/math8071108.

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Many problems in medical image reconstruction and machine learning can be formulated as nonconvex set theoretic feasibility problems. Among efficient methods that can be put to work in practice, successive projection algorithms have received a lot of attention in the case of convex constraint sets. In the present work, we provide a theoretical study of a general projection method in the case where the constraint sets are nonconvex and satisfy some other structural properties. We apply our algorithm to image recovery in magnetic resonance imaging (MRI) and to a signal denoising in the spirit of Cadzow’s method.

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5

FLETCHER, JAMES, and WARREN B. MOORS. "CHEBYSHEV SETS." Journal of the Australian Mathematical Society 98, no. 2 (November 11, 2014): 161–231. http://dx.doi.org/10.1017/s1446788714000561.

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AbstractA Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

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6

Vinter, Richard B. "The Hamiltonian Inclusion for Nonconvex Velocity Sets." SIAM Journal on Control and Optimization 52, no. 2 (January 2014): 1237–50. http://dx.doi.org/10.1137/130917417.

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7

Park, S., and I. S. Kim. "Remarks on saddle points in nonconvex sets." Applied Mathematics Letters 13, no. 1 (January 2000): 111–13. http://dx.doi.org/10.1016/s0893-9659(99)00153-6.

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8

Drusvyatskiy, D., A. D. Ioffe, and A. S. Lewis. "Transversality and Alternating Projections for Nonconvex Sets." Foundations of Computational Mathematics 15, no. 6 (August 21, 2015): 1637–51. http://dx.doi.org/10.1007/s10208-015-9279-3.

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9

Clarke, F. H., Yu S. Ledyaev, and R. J. Stern. "Fixed points and equilibria in nonconvex sets." Nonlinear Analysis: Theory, Methods & Applications 25, no. 2 (July 1995): 145–61. http://dx.doi.org/10.1016/0362-546x(94)00215-4.

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10

Li, Guoyin, Chunming Tang, Gaohang Yu, and Zengxin Wei. "On a Separation Principle for Nonconvex Sets." Set-Valued Analysis 16, no. 7-8 (July 12, 2008): 851–60. http://dx.doi.org/10.1007/s11228-008-0099-3.

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11

Ghafari, N., and H. Mohebi. "Optimality conditions for nonconvex problems over nearly convex feasible sets." Arabian Journal of Mathematics 10, no. 2 (March 2, 2021): 395–408. http://dx.doi.org/10.1007/s40065-021-00315-3.

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AbstractIn this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

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12

Wu, Congxin, Lixin Cheng, Minghu Ha, and E. S. Lee. "Convexification of nonconvex functions and application to minimum and maximum principles for nonconvex sets." Computers & Mathematics with Applications 31, no. 7 (April 1996): 27–36. http://dx.doi.org/10.1016/0898-1221(96)00016-8.

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13

Chen, Chi-Ming, Tong-Huei Chang, and Chiao-Wei Chung. "COINCIDENCE THEOREMS ON NONCONVEX SETS AND ITS APPLICATIONS." Taiwanese Journal of Mathematics 13, no. 2A (April 2009): 501–13. http://dx.doi.org/10.11650/twjm/1500405352.

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14

Lutwak, Erwin, Deane Yang, and Gaoyong Zhang. "The Brunn–Minkowski–Firey inequality for nonconvex sets." Advances in Applied Mathematics 48, no. 2 (February 2012): 407–13. http://dx.doi.org/10.1016/j.aam.2011.11.003.

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15

Nedić, Angelia, and Asuman Ozdaglar. "Separation of Nonconvex Sets with General Augmenting Functions." Mathematics of Operations Research 33, no. 3 (August 2008): 587–605. http://dx.doi.org/10.1287/moor.1070.0296.

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16

Ng, Kung Fu, and Rui Zang. "Linear regularity and ϕ-regularity of nonconvex sets." Journal of Mathematical Analysis and Applications 328, no. 1 (April 2007): 257–80. http://dx.doi.org/10.1016/j.jmaa.2006.05.028.

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17

The Luc, Dinh. "On the properly efficient points of nonconvex sets." European Journal of Operational Research 86, no. 2 (October 1995): 332–36. http://dx.doi.org/10.1016/0377-2217(94)00110-x.

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18

De Blasi, F. S., and G. Pianigiani. "On the Solution Sets of Nonconvex Differential Inclusions." Journal of Differential Equations 128, no. 2 (July 1996): 541–55. http://dx.doi.org/10.1006/jdeq.1996.0105.

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19

Li, Chong, and K. F. Ng. "On Best Approximation by Nonconvex Sets and Perturbation of Nonconvex Inequality Systems in Hilbert Spaces." SIAM Journal on Optimization 13, no. 3 (January 2002): 726–44. http://dx.doi.org/10.1137/s1052623402401373.

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20

Migórski, Stanisław, and Long Fengzhen. "Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets." Mathematics 8, no. 10 (October 17, 2020): 1824. http://dx.doi.org/10.3390/math8101824.

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In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

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21

Brykalov, S. A. "SOME PROPERTIES OF DIFFERENTIAL GAMES WITH NONCONVEX TARGET SETS." IFAC Proceedings Volumes 35, no. 1 (2002): 165–70. http://dx.doi.org/10.3182/20020721-6-es-1901.00596.

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22

Stéphane, Chrétien, and Bondon Pascal. "Cyclic projection methods on a class of nonconvex sets." Numerical Functional Analysis and Optimization 17, no. 1-2 (January 1996): 37–56. http://dx.doi.org/10.1080/01630569608816681.

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23

Ivanov, G. E., and M. S. Lopushanski. "A Separation Theorem for Nonconvex Sets and its Applications." Journal of Mathematical Sciences 245, no. 2 (January 23, 2020): 125–54. http://dx.doi.org/10.1007/s10958-020-04683-7.

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24

Kasimbeyli, Refail, and Masoud Karimi. "Separation theorems for nonconvex sets and application in optimization." Operations Research Letters 47, no. 6 (November 2019): 569–73. http://dx.doi.org/10.1016/j.orl.2019.09.011.

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25

Colombo, Giovanni, and Khai T. Nguyen. "Quantitative isoperimetric inequalities for a class of nonconvex sets." Calculus of Variations and Partial Differential Equations 37, no. 1-2 (June 30, 2009): 141–66. http://dx.doi.org/10.1007/s00526-009-0256-z.

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26

Brykalov, S. A. "Continuous strategies in differential games with nonconvex target sets." Doklady Mathematics 73, no. 2 (June 2006): 207–9. http://dx.doi.org/10.1134/s1064562406020141.

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27

Keller, André A. "Convex underestimating relaxation techniques for nonconvex polynomial programming problems: computational overview." Journal of the Mechanical Behavior of Materials 24, no. 3-4 (August 1, 2015): 129–43. http://dx.doi.org/10.1515/jmbm-2015-0015.

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AbstractThis paper introduces constructing convex-relaxed programs for nonconvex optimization problems. Branch-and-bound algorithms are convex-relaxation-based techniques. The convex envelopes are important, as they represent the uniformly best convex underestimators for nonconvex polynomials over some region. The reformulation-linearization technique (RLT) generates linear programming (LP) relaxations of a quadratic problem. RLT operates in two steps: a reformulation step and a linearization (or convexification) step. In the reformulation phase, the constraint and bound inequalities are replaced by new numerous pairwise products of the constraints. In the linearization phase, each distinct quadratic term is replaced by a single new RLT variable. This RLT process produces an LP relaxation. The LP-RLT yieds a lower bound on the global minimum. LMI formulations (linear matrix inequalities) have been proposed to treat efficiently with nonconvex sets. An LMI is equivalent to a system of polynomial inequalities. A semialgebraic convex set describes the system. The feasible sets are spectrahedra with curved faces, contrary to the LP case with polyhedra. Successive LMI relaxations of increasing size yield the global optimum. Nonlinear inequalities are converted to an LMI form using Schur complements. Optimizing a nonconvex polynomial is equivalent to the LP over a convex set. Engineering application interests include system analysis, control theory, combinatorial optimization, statistics, and structural design optimization.

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28

Antczak, Tadeusz. "Saddle point criteria in semi-infinite minimax fractional programming under (Φ,ρ)-invexity." Filomat 31, no. 9 (2017): 2557–74. http://dx.doi.org/10.2298/fil1709557a.

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Semi-infinite minimax fractional programming problems with both inequality and equality constraints are considered. The sets of parametric saddle point conditions are established for a new class of nonconvex differentiable semi-infinite minimax fractional programming problems under(?,?)-invexity assumptions. With the reference to the said concept of generalized convexity, we extend some results of saddle point criteria for a larger class of nonconvex semi-infinite minimax fractional programming problems in comparison to those ones previously established in the literature.

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29

Su, Menglong, Yufeng Shang, and Wenzhuang Zhu. "A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints." Mathematical Problems in Engineering 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/4067202.

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In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in more general nonconvex sets. Under suitable conditions, we obtain the global convergence of this homotopy continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper.

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30

SHAHBEYK, SHOKOUH, and MAJID SOLEIMANI-DAMANEH. "Limiting proper minimal points of nonconvex sets in finite-dimensional spaces." Carpathian Journal of Mathematics 35, no. 3 (2019): 379–84. http://dx.doi.org/10.37193/cjm.2019.03.12.

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In this paper, limiting proper minimal points of nonconvex sets in Euclidean finite-dimensional spaces are investigated. The relationships between these minimal points and Borwein, Benson, and Henig proper minimal points, under appropriate assumptions, are established. Furthermore, a density property is derived and a linear characterization of limiting proper minimal points is provided.

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31

Kaczor, Wieslawa. "Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets." Abstract and Applied Analysis 2003, no. 2 (2003): 83–91. http://dx.doi.org/10.1155/s1085337503205054.

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It is shown that ifXis a Banach space andCis a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets{Ci:1≤i≤n }ofX, and eachCihas the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping ofChas a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.

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32

Phu, H. X. "Some properties of solution sets to nonconvex quadratic programming problems." Optimization 56, no. 3 (June 2007): 369–83. http://dx.doi.org/10.1080/02331930600819597.

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33

De Blasi, Francesco S., and Giulio Pianigiani. "Solution sets of boundary value problems for nonconvex differential inclusions." Topological Methods in Nonlinear Analysis 1, no. 2 (June 1, 1993): 303. http://dx.doi.org/10.12775/tmna.1993.022.

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34

Kojima, Masakazu, and Akiko Takeda. "Complexity Analysis of Successive Convex Relaxation Methods for Nonconvex Sets." Mathematics of Operations Research 26, no. 3 (August 2001): 519–42. http://dx.doi.org/10.1287/moor.26.3.519.10580.

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35

Bienstock, Daniel, and Alexander Michalka. "Cutting-Planes for Optimization of Convex Functions over Nonconvex Sets." SIAM Journal on Optimization 24, no. 2 (January 2014): 643–77. http://dx.doi.org/10.1137/120878963.

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36

Kaczor, Wiesława. "Fixed points of λ-firmly nonexpansive mappings on nonconvex sets." Nonlinear Analysis: Theory, Methods & Applications 47, no. 4 (August 2001): 2787–92. http://dx.doi.org/10.1016/s0362-546x(01)00397-2.

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37

Alghamdi, Maryam A., Donal O'Regan, and Naseer Shahzad. "Krasnosel’skii Type Fixed Point Theorems for Mappings on Nonconvex Sets." Abstract and Applied Analysis 2012 (2012): 1–23. http://dx.doi.org/10.1155/2012/267531.

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We prove Krasnosel'skii type fixed point theorems in situations where the domain is not necessarily convex. As an application, the existence of solutions for perturbed integral equation is considered inp-normed spaces.

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38

Ubhaya, Vasant A. "Lipschitzian selections in approximation from nonconvex sets of bounded functions." Journal of Approximation Theory 56, no. 2 (February 1989): 217–24. http://dx.doi.org/10.1016/0021-9045(89)90111-1.

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39

Bogetoft, Peter, Joseph M. Tama, and Jørgen Tind. "Convex Input and Output Projections of Nonconvex Production Possibility Sets." Management Science 46, no. 6 (June 2000): 858–69. http://dx.doi.org/10.1287/mnsc.46.6.858.11938.

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40

Kojima, Masakazu, and Levent Tunçel. "Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets." SIAM Journal on Optimization 10, no. 3 (January 2000): 750–78. http://dx.doi.org/10.1137/s1052623498336450.

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41

Bauschke, Heinz H., Hung M. Phan, and Xianfu Wang. "The Method of Alternating Relaxed Projections for Two Nonconvex Sets." Vietnam Journal of Mathematics 42, no. 4 (December 31, 2013): 421–50. http://dx.doi.org/10.1007/s10013-013-0049-8.

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42

Drusvyatskiy, D., and A. S. Lewis. "Local Linear Convergence for Inexact Alternating Projections on Nonconvex Sets." Vietnam Journal of Mathematics 47, no. 3 (July 31, 2019): 669–81. http://dx.doi.org/10.1007/s10013-019-00357-3.

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43

Bonnisseau, Jean-Marc. "Existence of Lindahl equilibria in economies with nonconvex production sets." Journal of Economic Theory 54, no. 2 (August 1991): 409–16. http://dx.doi.org/10.1016/0022-0531(91)90131-m.

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44

BAROV, STOYU, and JAN J. DIJKSTRA. "ON CLOSED SETS IN HILBERT SPACE WITH CONVEX PROJECTIONS UNDER SOMEWHERE DENSE SETS OF DIRECTIONS." Journal of Topology and Analysis 02, no. 01 (March 2010): 123–43. http://dx.doi.org/10.1142/s1793525310000252.

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Let k be a fixed natural number. In an earlier paper the authors show that if C is a closed and nonconvex set in the Hilbert space ℓ2 such that the closures of the projections onto allk-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of ℓ2. Here this theorem is strengthened significantly by making the much weaker assumption that the set of projection directions is somewhere dense. To show the sharpness of the main theorem we construct "minimal imitations" of closed convex sets in ℓ2. In addition, we show that closed convex sets with an empty geometric interior cannot be imitated by other closed sets.

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45

Chen, Jinzuo, Ariana Pitea, and Li-Jun Zhu. "Split Systems of Nonconvex Variational Inequalities and Fixed Point Problems on Uniformly Prox-Regular Sets." Symmetry 11, no. 10 (October 12, 2019): 1279. http://dx.doi.org/10.3390/sym11101279.

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In this paper, we studied variational inequalities and fixed point problems in nonconvex cases. By the projection method over prox-regularity sets, the convergence of the suggested iterative scheme was established under some mild rules.

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46

Bounkhel, Messaoud. "Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces." Journal of Function Spaces 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/478437.

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The present paper is devoted to the study of the generalized projectionπK:X∗→K, whereXis a uniformly convex and uniformly smooth Banach space andKis a nonempty closed (not necessarily convex) set inX. Our main result is the density of the pointsx∗∈X∗having unique generalized projection over nonempty close sets inX. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.

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47

Khan, Abdul Rahim, A. B. Thaheem, and Nawab Hussain. "Random Fixed Points and Random Approximations in Nonconvex Domains." Journal of Applied Mathematics and Stochastic Analysis 15, no. 3 (January 1, 2002): 247–53. http://dx.doi.org/10.1155/s1048953302000217.

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Stochastic generalizations of some fixed point theorems on a class of nonconvex sets in a locally bounded topological vector space are established. As applications, Brosowski-Meinardus type theorems about random invariant approximation are obtained. This work extends or provides stochastic versions of several well known results.

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48

Zhao, Wenling, Daojin Song, and Bingzhuang Liu. "Error Bounds and Finite Termination for Constrained Optimization Problems." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/158780.

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We present a global error bound for the projected gradient of nonconvex constrained optimization problems and a local error bound for the distance from a feasible solution to the optimal solution set of convex constrained optimization problems, by using the merit function involved in the sequential quadratic programming (SQP) method. For the solution sets (stationary points set andKKTpoints set) of nonconvex constrained optimization problems, we establish the definitions of generalized nondegeneration and generalized weak sharp minima. Based on the above, the necessary and sufficient conditions for a feasible solution of the nonconvex constrained optimization problems to terminate finitely at the two solutions are given, respectively. Accordingly, the results in this paper improve and popularize existing results known in the literature. Further, we utilize the global error bound for the projected gradient with the merit function being computed easily to describe these necessary and sufficient conditions.

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49

Phan, Duy Nhat, Hoai An Le Thi, and Tao Pham Dinh. "Sparse Covariance Matrix Estimation by DCA-Based Algorithms." Neural Computation 29, no. 11 (November 2017): 3040–77. http://dx.doi.org/10.1162/neco_a_01012.

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This letter proposes a novel approach using the [Formula: see text]-norm regularization for the sparse covariance matrix estimation (SCME) problem. The objective function of SCME problem is composed of a nonconvex part and the [Formula: see text] term, which is discontinuous and difficult to tackle. Appropriate DC (difference of convex functions) approximations of [Formula: see text]-norm are used that result in approximation SCME problems that are still nonconvex. DC programming and DCA (DC algorithm), powerful tools in nonconvex programming framework, are investigated. Two DC formulations are proposed and corresponding DCA schemes developed. Two applications of the SCME problem that are considered are classification via sparse quadratic discriminant analysis and portfolio optimization. A careful empirical experiment is performed through simulated and real data sets to study the performance of the proposed algorithms. Numerical results showed their efficiency and their superiority compared with seven state-of-the-art methods.

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50

Nguyen, Luong V., and Xiaolong Qin. "The minimal time function associated with a collection of sets." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 93. http://dx.doi.org/10.1051/cocv/2020017.

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We define the minimal time function associated with a collection of sets which is motivated by the optimal time problem for nonconvex constant dynamics. We first provide various basic properties of this new function: lower semicontinuity, principle of optimality, convexity, Lipschitz continuity, among others. We also compute and estimate proximal, Fréchet and limiting subdifferentials of the new function at points inside the target set as well as at points outside the target. An application to location problems is also given.

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