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Variational analysis of circular cone programs

OPTIMIZATION 2015年第1期64卷 113-147页

作者： Zhou, Jinchuan Chen, Jein-Shah Mordukhovich, Boris S. Shandong Univ Technol Sch Sci Dept Math Zibo Peoples R China Natl Taiwan Normal Univ Dept Math Taipei Taiwan Wayne State Univ Dept Math Detroit MI 48202 USA King Fahd Univ Petr & Minerals Dhahran 31261 Saudi Arabia

This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming, important for optimization theory and its applications. First, we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/co-derivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs.

关键词： 49J52 90C30 90C31 generalized differentiation conic programming projection operator full and tilt stability circular cone variational analysis second-order cone optimization

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Generalized Farkas Lemma with Adjustable Variables and Two-Stage Robust Linear Programs

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2020年第2期187卷 488-519页

作者： Chuong, Thai Doan Jeyakumar, Vaithilingam Ton Duc Thang Univ Optimizat & Applicat Res Grp Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Math & Stat Ho Chi Minh City Vietnam Univ New South Wales Dept Appl Math Sydney NSW 2052 Australia

In this paper, we establish strong duality between affinely adjustable two-stage robust linear programs and their dual semidefinite programs under a general uncertainty set, that covers most of the commonly used uncertainty sets of robust optimization. This is achieved by first deriving a new version of Farkas' lemma for a parametric linear inequality system with affinely adjustable variables. Our strong duality theorem not only shows that the primal and dual program values are equal, but also allows one to find the value of a two-stage robust linear program by solving a semidefinite linear program. In the case of an ellipsoidal uncertainty set, it yields a corresponding strong duality result with a second-order cone program as its dual. To illustrate the efficacy of our results, we show how optimal storage cost of an adjustable two-stage lot-sizing problem under a ball uncertainty set can be found by solving its dual semidefinite program, using a commonly available software.

关键词： Generalized Farkas' lemma Semi-infinite linear system Adjustable robust linear programming Strong duality conic programming

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Dualize it: software for automatic primal and dual conversions of conic programs

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OPTIMIZATION METHODS & SOFTWARE 2009年第3期24卷 313-325页

作者： Lofberg, Johan Linkoping Univ Dept Elect Engn Div Automat Control S-58183 Linkoping Sweden

Many optimization problems gain from being interpreted and solved in either primal or dual forms. For a user with a particular application, one of these forms is usually much more natural to use, but this is not always the most efficient. This paper presents an implementation in the optimization modelling tool YALMIP that allows the user to define conic optimization problems in a preferred format, and then automatically derive a symbolic YALMIP model of the dual of this problem, solve the dual, and recover original variables. Applications in flexible generation of sum-of-squares programs, and efficient formulations of large-scale experiment design problems are used as illustrative examples.

关键词： optimization conic programming modelling software

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Optimality Conditions in DC-Constrained Mathematical programming Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第3期198卷 1191-1225页

作者： Correa, Rafael Lopez, Marco A. Perez-Aros, Pedro Univ OHiggins Rancagua Chile Univ Chile DIM CMM Santiago Chile Univ Alicante Dept Math Alicante 03071 Spain Federat Univ Australia Ctr Informat & Appl Optimizat Ballarat Australia Univ OHiggins Inst Ciencias Ingn Rancagua Chile

This paper provides necessary and sufficient optimality conditions for abstract-constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections.

关键词： DC functions DC-constrained programming conic programming Infinite programming Stochastic programming Semi-definite programming Supremum function

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Autonomous and conventional bus fleet optimization for fixed-route operations considering demand uncertainty

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TRANSPORTATION 2021年第5期48卷 2735-2763页

作者： Tian, Qingyun Lin, Yun Hui Wang, David Z. W. Nanyang Technol Univ Sch Civil & Environm Engn 50 Nanyang Ave Singapore 639798 Singapore Natl Univ Singapore Dept Ind Syst Engn & Management 1 Engn Dr 2Blk E1A 06-25 Singapore 117576 Singapore

The emerging technology of autonomous vehicles has been widely recognized as a promising urban mobility solution in the future. This paper considers the integration of autonomous vehicles into bus transit systems and proposes a modeling framework to determine the optimal bus fleet size and its assignment onto multiple bus lines in a bus service network considering uncertain demand. The mixed-integer stochastic programming approach is applied to formulate the problem. We apply the sample average approximation (SAA) method to solve the formulated stochastic programming problem. To tackle the nonconvexity of the SAA problem, we first present a reformulation method that transforms the problem into a mixed-integer conic quadratic program (MICQP), which can be solved to its global optimal solution by using some existing solution methods. However, this MICQP based approach can only handle the small-size problems. For the cases with large problem size, we apply the approach of quadratic transform with linear alternating algorithm, which allows for efficient solution to large-scale instances with up to thousands of scenarios in a reasonable computational time. Numerical results demonstrate the benefits of introducing autonomous buses as they are flexible to be assigned across different bus service lines, especially when demand uncertainty is more significant. The introduction of autonomous buses would enable further reduction of the required fleets and total cost. The model formulation and solution methods proposed in this study can be used to provide bus transit operators with operational guidance on including autonomous buses into bus services, especially on the autonomous and conventional bus fleets composition and allocation.

关键词： Autonomous bus service Demand uncertainty Stochastic programming conic programming Quadratic transform

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CBLIB 2014: a benchmark library for conic mixed-integer and continuous optimization

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MATHEMATICAL programming COMPUTATION 2016年第2期8卷 191-214页

作者： Friberg, Henrik A. Tech Univ Denmark Dept Wind Energy Roskilde Denmark MOSEK ApS Copenhagen Denmark

The conic Benchmark Library is an ongoing community-driven project aiming to challenge commercial and open source solvers on mainstream cone support. In this paper, 121 mixed-integer and continuous second-order cone problem instances have been selected from 11 categories as representative for the instances available online. Since current file formats were found incapable, we embrace the new conic Benchmark Format as standard for conic optimization. Tools are provided to aid integration of this format with other software packages.

关键词： Problem instances conic programming Mixed integer programming

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Numerical limit analysis and plasticity criterion of a porous Coulomb material with elliptic cylindrical voids

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COMPTES RENDUS MECANIQUE 2015年第3期343卷 199-209页

作者： Pastor, Franck Pastor, Joseph Kondo, Djimedo Athenee Royal Victor Horta Brussels Belgium Univ Savoie CNRS UMR 5271 Lab LOCIE F-73376 Le Bourget Du Lac France UPMC Inst Jean Le Rond DAlembert CNRS UMR 7190 F-75252 Paris 05 France

The paper is devoted to a numerical Limit Analysis of a hollow cylindrical model with a Coulomb solid matrix (of confocal boundaries) considered in the case of a generalized plane strain. To this end, the static approach of Pastor et al. (2008) [18] for Drucker-Prager materials is first extended to Coulomb problems. A new mixed but rigorously kinematic-code is elaborated for Coulomb problems in the present case of symmetry, resulting also in a conic programming approach. Owing to the good conditioning of the resulting optimization problems, both methods give very close bounds by allowing highly refined meshes, as verified by comparing to existing exact solutions. In a second part, using the identity of Tresca (as special case of Coulomb) and von Mises materials in plane strain, the codes are used to assess the corresponding results of Mariani and Corigliano (2001) [13] and of Madou and Leblond (2012) [11] for circular and elliptic cylindrical voids in a von Mises matrix. Finally, the Coulomb problem is investigated, also in terms of projections on the coordinate planes of the principal macroscopic stresses. (C) 2015 Published by Elsevier Masson SAS on behalf of Academie des sciences.

关键词： Gurson-type models Cylindrical voids Porous Coulomb material Micromechanics Limit analysis Static and mixed methods conic programming

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Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker-Prager materials with spheroidal voids

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COMPTES RENDUS MECANIQUE 2014年第2期342卷 96-105页

作者： Pastor, Franck Kondo, Djimedo Athenee Royal Victor Horta Brussels Belgium UPMC CNRS UMR 7190 Inst Jean Le Rond dAlembert F-75252 Paris 05 France

The paper is devoted to a numerical Limit Analysis of a hollow spheroidal model with a Drucker-Prager solid matrix, for several values of the corresponding friction angle phi. In the first part of this study, the static and the mixed kinematic 3D-codes recently evaluated in [1] are modified to use the geometry defined in [2] for spheroidal cavities in the context of a von Mises matrix. The results in terms of macroscopic criteria are satisfactory for low and medium values of phi, but not enough for phi = 30 in the highly compressive part of the criterion. To improve these results, an original mixed approach, dedicated to the axisymmetric case, was elaborated with a specific discontinuous quadratic velocity field associated with the triangular finite element. Despite the less good conditioning inherent to the axisymmetric modelization, the resulting conic programming problem appears quite efficient, allowing one take into account numerical discretization refinements unreachable with the corresponding 3D mixed code. After a first validation in the case of spherical cavities whose exact solution is known, the final results for spheroidal voids are given for three usual values of the friction angle and two values of the cavity aspect factor. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

关键词： Gurson-type models Spheroidal voids Micromechanics Limit analysis Upper and lower bounds conic programming

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Adaptive computable approximation to cones of nonnegative quadratic functions

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OPTIMIZATION 2014年第6期63卷 955-980页

作者： Lu, Cheng Jin, Qingwei Fang, Shu-Cherng Wang, Zhenbo Xing, Wenxun Tsinghua Univ Dept Elect Engn Beijing 100084 Peoples R China Zhejiang Univ Dept Management Sci & Engn Hangzhou 310003 Zhejiang Peoples R China N Carolina State Univ Dept Ind & Syst Engn Raleigh NC 27695 USA Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China

Cones of nonnegative quadratic functions are keys to the understanding of quadratic optimization problems, since any quadratically constrained quadratic programming problem can be reformulated as a linear conic programming problem over such a cone. This paper proposes an adaptive computable approximation scheme to cones of nonnegative quadratic functions and uses it for solving linear conic programming problems over such a cone. We study some basic properties of cones of nonnegative quadratic functions and present a class of simple cones with computable linear matrix inequalities representations. Building on these simple cones, we design a computable approximation scheme for handling a general cone of nonnegative quadratic functions. When the scheme is applied for solving linear conic programming problems over a cone of nonnegative quadratic functions, we incorporate an adaptive approach to enhance the performance of the proposed computable approximation scheme. The computational performance and theoretic convergence proof of the proposed adaptive computable approximation scheme are shown for solving box-constrained quadratic programming problems.

关键词： conic programming quadratic programming cones of nonnegative quadratic functions conic approximation lower bounds 49N15 49M37 90C26 90C20

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Topology optimization of truss structures under failure probability using the Bernstein approximation

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COMPUTERS & STRUCTURES 2024年 296卷

作者： Canelas, Alfredo Carrasco, Miguel Lopez, Julio Univ Republica Fac Ingn Julio Herrera & Reissig 565 Montevideo 11300 Uruguay Univ Andes Fac Ingn & Ciencias Aplicadas Mons Alvaro Portillo 12455 Santiago Chile Univ Diego Portales Fac Ingn & Ciencias Avda Ejercito 441 Santiago Chile

A novel topology optimization approach for the robust design of structures is presented. The method considers both deterministic and random loadings, and minimizes the compliance subject to a constraint on the volume, as well as a constraint on the failure probability. Handling the failure probability is often challenging in numerical terms, potentially leading to an intractable model as the problem scales. It is addressed by employing the Bernstein approximation, resulting in a model that has the remarkable property of being a linear conic programming problem, therefore, solvable in polynomial time with respect to the input size by using interior point methods. Furthermore, a more efficient reformulation of the problem, involving small semidefinite constraints is derived. To demonstrate the practicality of the proposed method, solutions to several examples of truss topology optimization are provided.

关键词： Robust optimization Reliability design optimization Topology optimization Bernstein approximation conic programming

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