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Journal articles on the topic 'Infinite-Dimensional linear programming'

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Published: 1 June 2024

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1

Appa, Gautam, Edward J. Anderson, and Peter Nash. "Linear Programming in Infinite-Dimensional Spaces." Journal of the Operational Research Society 40, no. 1 (January 1989): 109. http://dx.doi.org/10.2307/2583085.

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2

Appa, Gautam. "Linear Programming in Infinite-Dimensional Spaces." Journal of the Operational Research Society 40, no. 1 (January 1989): 109–10. http://dx.doi.org/10.1057/jors.1989.13.

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3

Romeijn, H. Edwin, Robert L. Smith, and James C. Bean. "Duality in infinite dimensional linear programming." Mathematical Programming 53, no. 1-3 (January 1992): 79–97. http://dx.doi.org/10.1007/bf01585695.

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4

López, M. A. "Linear programming in infinite-dimensional spaces." European Journal of Operational Research 36, no. 1 (July 1988): 134–35. http://dx.doi.org/10.1016/0377-2217(88)90019-7.

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5

Romeijn, H. Edwin, and Robert L. Smith. "Shadow Prices in Infinite-Dimensional Linear Programming." Mathematics of Operations Research 23, no. 1 (February 1998): 239–56. http://dx.doi.org/10.1287/moor.23.1.239.

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6

Ho, Tvu-Ying, Yuung-Yih Lur, and Soon-Yi Wu. "The Difference between Finite Dimensional Linear Programming Problems and Infinite Dimensional Linear Programming Problems." Journal of Mathematical Analysis and Applications 207, no. 1 (March 1997): 192–205. http://dx.doi.org/10.1006/jmaa.1997.5279.

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7

Taksar, Michael I. "Infinite-Dimensional Linear Programming Approach to SingularStochastic Control." SIAM Journal on Control and Optimization 35, no. 2 (March 1997): 604–25. http://dx.doi.org/10.1137/s036301299528685x.

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8

Vinh, N. T., D. S. Kim, N. N. Tam, and N. D. Yen. "Duality gap function in infinite dimensional linear programming." Journal of Mathematical Analysis and Applications 437, no. 1 (May 2016): 1–15. http://dx.doi.org/10.1016/j.jmaa.2015.12.043.

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9

Balbas, Alejandro, and Antonio Heras. "Duality theory for infinite-dimensional multiobjective linear programming." European Journal of Operational Research 68, no. 3 (August 1993): 379–88. http://dx.doi.org/10.1016/0377-2217(93)90194-r.

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10

Kariotoglou, Nikolaos, Maryam Kamgarpour, Tyler H. Summers, and John Lygeros. "The Linear Programming Approach to Reach-Avoid Problems for Markov Decision Processes." Journal of Artificial Intelligence Research 60 (October 4, 2017): 263–85. http://dx.doi.org/10.1613/jair.5500.

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One of the most fundamental problems in Markov decision processes is analysis and control synthesis for safety and reachability specifications. We consider the stochastic reach-avoid problem, in which the objective is to synthesize a control policy to maximize the probability of reaching a target set at a given time, while staying in a safe set at all prior times. We characterize the solution to this problem through an infinite dimensional linear program. We then develop a tractable approximation to the infinite dimensional linear program through finite dimensional approximations of the decision space and constraints. For a large class of Markov decision processes modeled by Gaussian mixtures kernels we show that through a proper selection of the finite dimensional space, one can further reduce the computational complexity of the resulting linear program. We validate the proposed method and analyze its potential with numerical case studies.
11

Ito, Satoshi, Soon-Yi Wu, Ting-Jang Shiu, and Kok Lay Teo. "A numerical approach to infinite-dimensional linear programming in $L_1$ spaces." Journal of Industrial & Management Optimization 6, no. 1 (2010): 15–28. http://dx.doi.org/10.3934/jimo.2010.6.15.

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12

Mendiondo, Marta Susana, and Richard H. Stockbridge. "Approximation of Infinite-Dimensional Linear Programming Problems which Arise in Stochastic Control." SIAM Journal on Control and Optimization 36, no. 4 (July 1998): 1448–72. http://dx.doi.org/10.1137/s0363012996313367.

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13

Mulvey, J. M. "Infinite programming: Proceedings of an international symposium on infinite dimensional linear programming, Churchill College, Cambridge, United Kingdom, September 7–10, 1984." European Journal of Operational Research 27, no. 2 (October 1986): 259. http://dx.doi.org/10.1016/0377-2217(86)90077-9.

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14

ERIKSSON, BJORN, and MARTIJN PISTORIUS. "METHOD OF MOMENTS APPROACH TO PRICING DOUBLE BARRIER CONTRACTS IN POLYNOMIAL JUMP-DIFFUSION MODELS." International Journal of Theoretical and Applied Finance 14, no. 07 (November 2011): 1139–58. http://dx.doi.org/10.1142/s0219024911006644.

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We present a method of moments approach to pricing double barrier contracts when the underlying is modelled by a polynomial jump-diffusion. By general principles the price is linked to certain infinite dimensional linear programming problems. Subsequently approximating these by finite dimensional linear programming problems, upper and lower bounds for the prices of such options are found. We derive theoretical convergence results for this algorithm, and provide numerical illustrations by applying the method to the valuation of several double barrier-type contracts (double barrier knock-out call, American corridor and double-no-touch options) under a number of different models, also allowing for a deterministic short rate.
15

Khanh, Pham Duy, Tran Hong Mo, and Trinh T. T. Tran. "Necessary and Sufficient Conditions for Qualitative Properties of Infinite Dimensional Linear Programming Problems." Numerical Functional Analysis and Optimization 40, no. 8 (March 16, 2019): 924–43. http://dx.doi.org/10.1080/01630563.2019.1566244.

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16

Levin, V. L. "On generic uniqueness of optimal solution in an infinite dimensional linear programming problem." Doklady Mathematics 78, no. 1 (August 2008): 490–92. http://dx.doi.org/10.1134/s1064562408040054.

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17

Nordebo, S., and Z. Zang. "Application of infinite dimensional linear programming to IIR filter design with time domain constraints." IEEE Transactions on Signal Processing 47, no. 7 (July 1999): 2060–63. http://dx.doi.org/10.1109/78.771057.

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18

Staffans, Olof J. "The Four-Block Model Matching Problem in $l^1 $ and Infinite-Dimensional Linear Programming." SIAM Journal on Control and Optimization 31, no. 3 (May 1993): 747–79. http://dx.doi.org/10.1137/0331034.

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19

Klabjan, Diego, and Daniel Adelman. "An Infinite-Dimensional Linear Programming Algorithm for Deterministic Semi-Markov Decision Processes on Borel Spaces." Mathematics of Operations Research 32, no. 3 (August 2007): 528–50. http://dx.doi.org/10.1287/moor.1070.0252.

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20

Bartl, David. "Farkas' Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach." Linear and Multilinear Algebra 55, no. 4 (July 2007): 327–53. http://dx.doi.org/10.1080/03081080600967820.

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21

Fang, Zheng, Andres Santos, Azeem M. Shaikh, and Alexander Torgovitsky. "Inference for Large‐Scale Linear Systems With Known Coefficients." Econometrica 91, no. 1 (2023): 299–327. http://dx.doi.org/10.3982/ecta18979.

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This paper considers the problem of testing whether there exists a non‐negative solution to a possibly under‐determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of settings, including random coefficient, treatment effect, and discrete choice models, as well as a class of linear programming problems. As a first contribution, we obtain a novel geometric characterization of the null hypothesis in terms of identified parameters satisfying an infinite set of inequality restrictions. Using this characterization, we devise a test that requires solving only linear programs for its implementation, and thus remains computationally feasible in the high‐dimensional applications that motivate our analysis. The asymptotic size of the proposed test is shown to equal at most the nominal level uniformly over a large class of distributions that permits the number of linear equations to grow with the sample size.
22

Boucekkine, Raouf, Giorgio Fabbri, Salvatore Federico, and Fausto Gozzi. "Geographic environmental Kuznets curves: the optimal growth linear-quadratic case." Mathematical Modelling of Natural Phenomena 14, no. 1 (2019): 105. http://dx.doi.org/10.1051/mmnp/2018076.

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We solve a linear-quadratic model of a spatio-temporal economy using a polluting one-input technology. Space is continuous and heterogenous: locations differ in productivity, nature self-cleaning technology and environmental awareness. The unique link between locations is transboundary pollution which is modelled as a PDE diffusion equation. The spatio-temporal functional is quadratic in local consumption and linear in pollution. Using a dynamic programming method adapted to our infinite dimensional setting, we solve the associated optimal control problem in closed-form and identify the asymptotic (optimal) spatial distribution of pollution. We show that optimal emissions will decrease at given location if and only if local productivity is larger than a threshold which depends both on the local pollution absorption capacity and environmental awareness. Furthermore, we numerically explore the relationship between the spatial optimal distributions of production and (asymptotic) pollution in order to uncover possible (geographic) environmental Kuznets curve cases.
23

Friedland, Shmuel, Jingtong Ge, and Lihong Zhi. "Quantum Strassen’s theorem." Infinite Dimensional Analysis, Quantum Probability and Related Topics 23, no. 03 (September 2020): 2050020. http://dx.doi.org/10.1142/s0219025720500204.

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Strassen’s theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite, Strassen’s theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability matrix on two finite product spaces. Partial traces of the density matrix are analogs of marginals. The support of the density matrix is its range. The analog of Strassen’s theorem in this case can be stated and solved using semidefinite programming. The aim of this paper is to give analogs of Strassen’s theorem to density trace class operators on a product of two separable Hilbert spaces, where at least one of the Hilbert spaces is infinite-dimensional.
24

Glimm, Tilmann, and Nick Henscheid. "Iterative Scheme for Solving Optimal Transportation Problems Arising in Reflector Design." ISRN Applied Mathematics 2013 (November 24, 2013): 1–12. http://dx.doi.org/10.1155/2013/635263.

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We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it has been shown previously that this problem is equivalent to an infinite-dimensional linear programming (LP) problem. Here we investigate techniques for constructing the two reflectors numerically by considering the finite-dimensional LP problems which arise as approximations to the infinite-dimensional problem. A straightforward discretization has the disadvantage that the number of constraints increases rapidly with the mesh size, so only very coarse meshes are practical. To address this well-known issue we propose an iterative solution scheme. In each step, an LP problem is solved, where information from the previous iteration step is used to reduce the number of necessary constraints. As an illustration, we apply our proposed scheme to solve a problem with synthetic data, demonstrating that the method allows for much finer meshes than a simple discretization. We also give evidence that the scheme converges. There exists a growing literature for the application of optimal transportation theory to other beam shaping problems, and our proposed scheme is easy to adapt for these problems as well.
25

Pattnaik, Monalisha. "Optimality test in fuzzy inventory model for restricted budget and space: Move forward to a non-linear programming approach." Yugoslav Journal of Operations Research 25, no. 3 (2015): 457–70. http://dx.doi.org/10.2298/yjor130517023p.

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In this paper, the concept of fuzzy Non-Linear Programming Technique is applied to solve an economic order quantity (EOQ) model for restricted budget and space. Since various types of uncertainties and imprecision are inherent in real inventory problems, they are classically modeled using the approaches from the probability theory. However, there are uncertainties that cannot be appropriately treated by the usual probabilistic models. The questions are how to define inventory optimization tasks in such environment and how to interpret the optimal solutions. This paper allow the modification of the Single item EOQ model in presence of fuzzy decision making process where demand is related to the unit price, and the setup cost varies with the quantity produced/Purchased. The modification of objective function, budget, and storage area in the presence of imprecisely estimated parameters are considered. The model is developed by employing different approaches over an infinite planning horizon. It incorporates all the concepts of a fuzzy arithmetic approach and comparative analysis with other non linear models. Investigation of the properties of an optimal solution allows developing an algorithm whose validity is illustrated by an example problem, and two and three dimensional diagrams are represented to this application through MATL(R2009a) software. Sensitivity analysis of the optimal solution is studied with respect to the changes of different parameter values for obtaining managerial insights of the decision problem.
26

Rusmevichientong, Paat, Mika Sumida, and Huseyin Topaloglu. "Dynamic Assortment Optimization for Reusable Products with Random Usage Durations." Management Science 66, no. 7 (July 2020): 2820–44. http://dx.doi.org/10.1287/mnsc.2019.3346.

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We consider dynamic assortment problems with reusable products, in which each arriving customer chooses a product within an offered assortment, uses the product for a random duration of time, and returns the product back to the firm to be used by other customers. The goal is to find a policy for deciding on the assortment to offer to each customer so that the total expected revenue over a finite selling horizon is maximized. The dynamic-programming formulation of this problem requires a high-dimensional state variable that keeps track of the on-hand product inventories, as well as the products that are currently in use. We present a tractable approach to compute a policy that is guaranteed to obtain at least 50% of the optimal total expected revenue. This policy is based on constructing linear approximations to the optimal value functions. When the usage duration is infinite or follows a negative binomial distribution, we also show how to efficiently perform rollout on a simple static policy. Performing rollout corresponds to using separable and nonlinear value function approximations. The resulting policy is also guaranteed to obtain at least 50% of the optimal total expected revenue. The special case of our model with infinite usage durations captures the case where the customers purchase the products outright without returning them at all. Under infinite usage durations, we give a variant of our rollout approach whose total expected revenue differs from the optimal by a factor that approaches 1 with rate cubic-root of Cmin, where Cmin is the smallest inventory of a product. We provide computational experiments based on simulated data for dynamic assortment management, as well as real parking transaction data for the city of Seattle. Our computational experiments demonstrate that the practical performance of our policies is substantially better than their performance guarantees and that performing rollout yields noticeable improvements. This paper was accepted by Yinyu Ye, optimization.
27

Larysa, Koriashkina, and Dziuba Serhii. "МАТЕМАТИЧНІ МОДЕЛІ ТА МЕТОДИ РОЗМІЩЕННЯ ОБ’ЄКТІВ І ЗОНУВАННЯ ТЕРИТОРІЙ В СИСТЕМАХ ЕКСТРЕНОЇ ЛОГІСТИКИ." System technologies 6, no. 149 (April 1, 2024): 107–22. http://dx.doi.org/10.34185/1562-9945-6-149-2023-09.

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The mathematical models for distribution processes related to organizing precautionary measures in the event of threats or occurrences of man-made emergencies are presented. The tasks include optimal zoning of territories with the fixing of zones by objects of social purpose for service provision. Provision is made for: the possibility of overlapping zones in case the nearest center cannot provide the service; optimal placement of a certain number of new cen-ters of emergency logistics systems with simultaneous redistribution of the load on all their structural elements; the selection of locations of structural subdivisions based on existing fa-cilities. The optimality criteria involve minimizing either the time to provide the service even to the most remote object in the given territory, or the total distance to the nearest centers from consumers that are densely distributed in the given territory, and/or the organizational costs associated with the arrangement of new centers. Mathematical models are proposed in the form of continuous problems of optimal multiplex partitioning of sets with a linear or minimax functional of quality. The latter provides such placement of centers that provides op-timal multiple coverage of the territory (with a minimum radius of multiple coverage). Meth-ods for solving the formulated problems were developed using LP-relaxation of linear prob-lems with Boolean variables, duality theory to reduce the initial problems of infinite-dimensional programming to problems of conditional optimization of a non-smooth function of several variables, and modern methods of non-differentiated optimization.
28

Malozemov, Vassili N., and Natalya A. Solovyeva. "MDM method for solving the general quadratic problem of mathematical diagnostics." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 10 (68), no. 3 (2023): 516–29. http://dx.doi.org/10.21638/spbu01.2023.306.

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The term mathematical diagnostics was introduced by V. F. Demyanov in the early 2000s. The simplest problem of mathematical diagnostics is to determine the relative position of a certain point p and the convex hull C of a finite number of given points in n-dimensional Euclidean space. Of interest is the answer to the following questions: does the point p belong to the set C or not? If p does not belong to C, then what is the distance from p to C? In general problem of mathematical diagnostics two convex hulls are considered. The question is whether they have common points. If there are no common points, then it is required to find the distance between these hulls. From an algorithmic point of view, the problems of mathematical diagnostics are reduced to special problems of linear or quadratic programming, for the solution of which there are finite methods. However, when implementing this approach in the case of large data arrays, serious computational difficulties arise. Infinite but easily implemented methods come to the rescue, which allow obtaining an approximate solution with the required accuracy in a finite number of iterations. These methods include the MDM method. It was developed by Mitchell, Demyanov and Malozemov in 1971 for other purposes, but later found application in machine learning. From a modern point of view, the original version of the MDM method can be used to solve the simplest problems of mathematical diagnostics. This article gives a natural generalization of the MDM-method, oriented towards solving general problems of mathematical diagnostics. The equivalence of the general problem of mathematical diagnostics and the problem of linear separation of two finite sets with the largest width of the margin is established.
29

Tovstik, Tatiana M. "Stationary reversibles processes MA and ARMA." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 10 (68), no. 3 (2023): 530–44. http://dx.doi.org/10.21638/spbu01.2023.307.

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Abstract:
The term mathematical diagnostics was introduced by V. F. Demyanov in the early 2000s. The simplest problem of mathematical diagnostics is to determine the relative position of a certain point p and the convex hull C of a finite number of given points in n-dimensional Euclidean space. Of interest is the answer to the following questions: does the point p belong to the set C or not? If p does not belong to C, then what is the distance from p to C? In general problem of mathematical diagnostics two convex hulls are considered. The question is whether they have common points. If there are no common points, then it is required to find the distance between these hulls. From an algorithmic point of view, the problems of mathematical diagnostics are reduced to special problems of linear or quadratic programming, for the solution of which there are finite methods. However, when implementing this approach in the case of large data arrays, serious computational difficulties arise. Infinite but easily implemented methods come to the rescue, which allow obtaining an approximate solution with the required accuracy in a finite number of iterations. These methods include the MDM method. It was developed by Mitchell, Demyanov and Malozemov in 1971 for other purposes, but later found application in machine learning. From a modern point of view, the original version of the MDM method can be used to solve the simplest problems of mathematical diagnostics. This article gives a natural generalization of the MDM-method, oriented towards solving general problems of mathematical diagnostics. The equivalence of the general problem of mathematical diagnostics and the problem of linear separation of two finite sets with the largest width of the margin is established.
30

Kamgarpour, Maryam, and Tyler Summers. "On infinite dimensional linear programming approach to stochastic control * *This research is partially supported by M. Kamgarpour’s European Union ERC Starting Grant, CONENE and by T. Summers’ the US National Science Foundation under grant CNS-1566127." IFAC-PapersOnLine 50, no. 1 (July 2017): 6148–53. http://dx.doi.org/10.1016/j.ifacol.2017.08.979.

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31

Alzalg, Baha. "Barrier methods based on Jordan-Hilbert algebras for stochastic optimization in spin factors." RAIRO - Operations Research, December 22, 2023. http://dx.doi.org/10.1051/ro/2023198.

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Infinite-dimensional stochastic second-order cone programming involves minimizing linear functions over intersections of affine linear manifolds with infinite-dimensional second-order cones. However, even though there is a legitimate necessity to explore these methods in general spaces, there is an absence of infinite-dimensional counterparts for these methods. In this paper, we present decomposition logarithmic-barrier interior-point methods based on unital Jordan-Hilbert algebras for this class of optimization problems in the infinite-dimensional setting. The results show that the iteration complexity of the proposed algorithms is independent on the choice of Hilbert spaces from which the underlying spin factors are formed, and so it coincides with the best-known complexity obtained by such methods for the finite-dimensional setting. We apply our results to an important problem in stochastic control, namely the two-stage stochastic multi-criteria design problem. We show that the corresponding infinite-dimensional system in this case is a matrix differential Ricatti equation plus a finite-dimensional system, and hence, it can be solved efficiently to find the search direction.
32

Leyffer, Sven, and Paul Manns. "Sequential linear integer programming for integer optimal control with total variation regularization." ESAIM: Control, Optimisation and Calculus of Variations, September 20, 2022. http://dx.doi.org/10.1051/cocv/2022059.

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We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty implies that the considered integer control problems admit minmizers. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example.
33

Zălinescu, Constantin. "On the duality gap and Gale's example in infinite-dimensional conic linear programming." Journal of Mathematical Analysis and Applications, November 2022, 126868. http://dx.doi.org/10.1016/j.jmaa.2022.126868.

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34

Rahaman, Mijanur, Mohd Ishtyak, Iqbal Ahmad, and Rais Ahmad. "Split monotone variational inclusion problem involving Cayley operators." Georgian Mathematical Journal, September 30, 2022. http://dx.doi.org/10.1515/gmj-2022-2187.

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Abstract In this article, we introduce a new kind of split monotone variational inclusion problem involving Cayley operator in the setting of infinite-dimensional Hilbert spaces. We develop a general iterative method to approximate the solution of the split monotone variational inclusion problem involving Cayley operator. Under some suitable conditions, a convergence theorem for the sequences generated by the proposed iterative scheme is established, which also solves certain variational inequality problems related to strongly positive linear operators. Finally, a numerical example is presented to study the efficiency of the proposed algorithm through MATLAB programming.
35

Milz, Johannes. "Sample average approximations of strongly convex stochastic programs in Hilbert spaces." Optimization Letters, May 28, 2022. http://dx.doi.org/10.1007/s11590-022-01888-4.

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AbstractWe analyze the tail behavior of solutions to sample average approximations (SAAs) of stochastic programs posed in Hilbert spaces. We require that the integrand be strongly convex with the same convexity parameter for each realization. Combined with a standard condition from the literature on stochastic programming, we establish non-asymptotic exponential tail bounds for the distance between the SAA solutions and the stochastic program’s solution, without assuming compactness of the feasible set. Our assumptions are verified on a class of infinite-dimensional optimization problems governed by affine-linear partial differential equations with random inputs. We present numerical results illustrating our theoretical findings.
36

Yu, Huizhen. "On Linear Programming for Constrained and Unconstrained Average-Cost Markov Decision Processes with Countable Action Spaces and Strictly Unbounded Costs." Mathematics of Operations Research, December 15, 2021. http://dx.doi.org/10.1287/moor.2021.1177.

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We consider the linear programming approach for constrained and unconstrained Markov decision processes (MDPs) under the long-run average-cost criterion, where the class of MDPs in our study have Borel state spaces and discrete countable action spaces. Under a strict unboundedness condition on the one-stage costs and a recently introduced majorization condition on the state transition stochastic kernel, we study infinite-dimensional linear programs for the average-cost MDPs and prove the absence of a duality gap and other optimality results. Our results do not require a lower-semicontinuous MDP model. Thus, they can be applied to countable action space MDPs where the dynamics and one-stage costs are discontinuous in the state variable. Our proofs make use of the continuity property of Borel measurable functions asserted by Lusin’s theorem.
37

Narciso, Diogo A. C., and Efstratios N. Pistikopoulos. "Novel solution strategies for multiparametric nonlinear optimization problems with convex objective function and linear constraints." Optimization and Engineering, April 22, 2024. http://dx.doi.org/10.1007/s11081-024-09888-2.

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AbstractThis paper expands the multiparametric quadratic programming (mp-QP) framework presented in Narciso et al. (Comput Chem Eng 164:107882, 2022. https://doi.org/10.1016/j.compchemeng.2022.107882) to the more general multiparametric nonlinear programming (mp-NLP) case. First, the vector of parameters in mp-NLP problems is recast so that a unique transformed parameter is implicitly assigned to each of the inequality constraints. Maps of critical regions in this transformed space of parameters feature a set of 1-dimensional parametric edges (two per inequality constraint), which then greatly facilitate solution calculation. In the mp-NLP case, however, parametric edges define nonlinear semi-infinite lines; this requires an adaptation to the mp-QP algorithm (deals with linear parametric edges only), to enable a suitable calculation path to the more general nonlinear case. Three routes are proposed to mp-NLPs: the first route delivers solutions in compact form (same format as in mp-QP) using a single reference point per edge; the second route delivers explicit solutions using a hybrid approach for critical region construction, where all active sets not detected in the parameters space are excluded from the solution (equivalent to first route concerning accuracy); the third route builds on the initial explicit solution and further partitions the parameters space until all solution fragments satisfy an error check. Five algorithms were coded for these routes, and tested in a large range of mp-NLP problems. These strategies enable significant improvements in terms of solution accuracy, algorithm efficiency, and interpretability when compared to the state-of-the-art mp-NLP algorithms.
38

Lim, Tongseok, and Robert J. McCann. "Geometrical Bounds for Variance and Recentered Moments." Mathematics of Operations Research, May 24, 2021. http://dx.doi.org/10.1287/moor.2021.1125.

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We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu and of Bhatia and Davis concerning measures on the line to several dimensions. This is done using convex duality and (infinite-dimensional) linear programming. The following consequence of our bounds exhibits symmetry breaking, provides a new proof of Jung’s theorem, and turns out to have applications to the aggregation dynamics modelling attractive–repulsive interactions: among probability measures on [Formula: see text] whose support has diameter at most [Formula: see text], we show that the variance around the mean is maximized precisely by those measures that assign mass [Formula: see text] to each vertex of a standard simplex. For [Formula: see text], the [Formula: see text] th moment—optimally centered—is maximized by the same measures among those satisfying the diameter constraint.
39

Zhang, Shixuan, and Xu Andy Sun. "Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization." Mathematical Programming, August 20, 2022. http://dx.doi.org/10.1007/s10107-022-01875-8.

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AbstractIn this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a $$(T+1)$$ ( T + 1 ) -stage stochastic MINLP satisfying L-exact Lipschitz regularization with d-dimensional state spaces, to obtain an $$\varepsilon $$ ε -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by $${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$ O ( ( 2 L T ε ) d ) , and is lower bounded by $${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$ O ( ( LT 4 ε ) d ) for the general case or by $${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$ O ( ( LT 8 ε ) d / 2 - 1 ) for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T, if all the state spaces are finite sets, or if we seek a $$(T\varepsilon )$$ ( T ε ) -optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.
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Fan, Xiangyi, and Grani A. Hanasusanto. "A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse." INFORMS Journal on Computing, November 29, 2023. http://dx.doi.org/10.1287/ijoc.2021.0306.

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Abstract:
We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0306 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0306 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

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