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Adjustable robust multiobjective linear optimization: Pareto optimal solutions via conic programming

ANNALS OF OPERATIONS RESEARCH 2025年第2期346卷 895-916页

作者： Chuong, T. D. Jeyakumar, V. Saigon Univ Dept Math & Applicat Ho Chi Minh City Vietnam RMIT Univ Sch Engn Melbourne Vic 3000 Australia Univ New South Wales Dept Appl Math Sydney 2052 Australia

In this paper we study two-stage affinely adjustable robust multi-objective optimization problems. We show how (weak) Pareto optimal solutions of these robust multi-objective problems can be found by solving conic linear programming problems. We do this by first deriving numerically verifiable conditions that characterize (weak) Pareto optimal solutions of affinely adjustable robust multi-objective programs under a spectrahedron uncertainty set. The uncertainty set covers most of the commonly used uncertainty sets of robust optimization. We then reformulate the weighted-sum optimization problems of the multi-objective problems, derived with the aid of the optimality conditions, as equivalent conic linear programming problems, such as semidefinite programs or second-order cone programs, to find the (weak) Pareto optimal solutions. We illustrate by an example how our results can be used to find a second-stage (weak) Pareto optimal solution by solving a semidefinite program using a commonly available software.

关键词： Multi-objective optimization Adjustable robust optimization Pareto optimal solution conic programming

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conic programming models for production planning with clearing functions: Formulations and duality

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2021年第3期292卷 953-966页

作者： Gopalswamy, Karthick Uzsoy, Reha North Carolina State Univ Edward P Fitts Dept Ind & Syst Engn Raleigh NC 27695 USA

Concave clearing functions that model the expected throughput of a production resource as a function of its planned workload have yielded promising results when used in production planning models. The most common of these models take the form of linear programs (LPs) obtained by piecewise linearization of the clearing function constraints, leading to large formulations and inaccurate estimates of dual prices for resources. We show that several clearing function forms considered in the literature to date can be reformulated as conic programs (CPs), for which efficient solution methods and an elegant duality theory analogous to that for linear programming exist. We derive expressions for the optimal values of the dual variables and present computational experiments showing that the dual prices obtained from the CP formulation are more accurate than those obtained from the piecewise linearized LP models. In addition, the CP solution outperforms the LP solutions with respect to nervousness. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Capacity pricing Congestion conic programming Production planning

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conic programming for Circuit Equations With Rational Current Controlled Resistors

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2021年第1期68卷 496-500页

作者： Jia, Wenhao Ding, Tao Qu, Ming Bai, Linquan Li, Fangxing Xi An Jiao Tong Univ Dept Elect Engn Xian 710049 Peoples R China Univ North Carolina Charlotte Syst Engn & Engn Management Charlotte NC 28223 USA Univ Tennessee Dept Elect Engn & Comp Sci Knoxville TN 37996 USA

This brief considers special nonlinear resistors are where the voltage is controlled by a polynomial of the current with arbitrary rational exponents, called rational current controlled resistors (RCCRs). It is discovered that nonlinear circuit equations with these special nonlinear resistors can be isomorphic to the Karush-Kuhn-Tucker (KKT) conditions of a convex optimization model. Moreover, three-dimensional rotated cones using the "tower of variables" technique are utilized to exactly reformulate the convex optimization model into tractable conic programming (CP). Through the proposed approach, the solution of the nonlinear circuit equations can be exactly obtained by solving the CP. Simulation results on several test systems with RCCRs demonstrate the effectiveness of the proposed method.

关键词： Manganese Mathematical model Resistors Nonlinear circuits Integrated circuit modeling Convex functions Convergence Rational current controlled resistor circuit equations conic programming second order cones

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An analytic center cutting plane approach for conic programming

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MATHEMATICS OF OPERATIONS RESEARCH 2008年第3期33卷 529-551页

作者： Basescu, Vasile L. Mitchell, John E. Campbell Co Towson MD 21204 USA Rensselaer Polytech Inst Dept Math Sci Troy NY 12180 USA

We analyze the problem of finding a point strictly interior to a bounded, convex, and fully dimensional set from a finite dimensional Hilbert space. We generalize the results obtained for the linear programming ( LP), semide finite programming (SDP), and second-order core programming (SOCP) cases. The cuts added by our algorithm are central and conic. In our analysis, we find an upper bound for the number of Newton steps required to compute an approximate analytic center. Also, we provide an upper bound for the total number of cuts added to solve the problem. This bound depends on the quality of the cuts, the dimensionality of the problem and the thickness of the set we are considering.

关键词： cutting plane cutting surface analytic center conic programming feasibility problem

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A conic programming approach to the wrinkling of pneumatic membranes using convex potentials

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COMPUTERS & STRUCTURES 2024年 292卷

作者： Niewiarowski, Alexander Adriaenssens, Sigrid Pauletti, Ruy Marcelo Princeton Univ Dept Civil & Environm Engn 59 Olden St Princeton NJ 08544 USA Univ Sao Paulo Polytech Sch Ave Prof Almeida PradoTrav 2 N 83 Sao Paulo SP Brazil

This paper presents a novel conic programming approach for modeling the wrinkling of isotropic hyperelastic membranes subject to nonlinear loading and finite strains, employing convex potentials derived from tension field theory. The incompressible neo-Hookean strain energy is recast as a minimization problem over a set of cones, with a semidefinite constraint on the deformed surface metric. To address pneumatic membranes, a linearized volumetric potential is introduced, reestablishing convexity for follower forces, and Boyle's law is expressed as a minimization problem over the exponential cone. The primal-dual interior point method solves the total potential energy minimization problem, with convergence verified using the analytical solution of a sheared planar membrane. The proposed model, applied to examples of increasing complexity, reveals intriguing membrane behaviors rarely discussed in existing literature. The method is shown to be robust, even when handling highly wrinkled membranes, which are challenging to address using direct nonlinear equilibrium analysis or nonlinear interior point methods. Implementation using high-level automatic code generation tools (FEniCS) results in concise, extensible code. The findings point to several avenues for future research, such as exploring complex material models, mesh refinement, dynamics, and parametric design. Additionally, the linearization process implies potential applicability of the methods to nonlinear problems beyond membrane mechanics.

关键词： Membrane mechanics Wrinkling Tension field theory conic programming Nonsmooth mechanics FEniCS

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A CONVERGENT 3-BLOCK SEMIPROXIMAL ALTERNATING DIRECTION METHOD OF MULTIPLIERS FOR conic programming WITH 4-TYPE CONSTRAINTS

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SIAM JOURNAL ON OPTIMIZATION 2015年第2期25卷 882-915页

作者： Sun, Defeng Toh, Kim-Chuan Yang, Liuqin Natl Univ Singapore Dept Math Singapore 119076 Singapore Natl Univ Singapore Risk Management Inst Singapore 119076 Singapore

In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order 1 -> 3 -> 2 -> 3, instead of the usual 1 -> 2 -> 3 Gauss-Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20% faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and/or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks.

关键词： conic programming multiblock ADMM semiproximal ADMM convergence SDP

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Numerical limit analysis of reinforced concrete slabs using a dual approach and conic programming

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2017年第4期55卷 1407-1423页

作者： Tinoco, Hector A. UAM Dept Mech & Prod Grp Diseno & Desarrollo Ind ARCHYTAS Antigua Estn Ferrocarril Edificio Sacatin Manizales Caldas 170001 Colombia

This study presents a numerical procedure for the analysis of reinforced concrete slabs (RCS) that obey Nielsen's yield criterion. A pseudo lower bound formulation of finite elements is established to solve the static theorem as a conic optimization problem with the aim to determine the maximum load capacity of RCS. Lower bound solution is improved by means of an adaptive remeshing strategy using a dissipation estimator which is obtained on the normalization of the kinematic criterion defined from the yield criteria. Different known examples are evaluated and the results show a good agreement with the exact values of the collapse load as well as numerical convergence. The displacement rates are estimated from the dual solution established in the optimization problem in order to determine the collapse mechanism in the slab.

关键词： conic programming Limit state analysis Shakedown theory Finite element method Reinforced concrete slabs Adaptive remeshing

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Applications of conic programming in Non-smooth Mechanics

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第1期202卷 340-372页

作者： Bleyer, Jeremy Univ Gustave Eiffel Lab Navier ENPC CNRS 6-8 Av Blaise Pascal F-77455 Marne La Vallee France

In the field of nonlinear mechanics, many challenging problems (e.g., plasticity, contact, masonry structures, nonlinear membranes) turn out to be expressible as conic programs. In general, such problems are non-smooth in nature (plasticity condition, unilateral condition, etc.), which makes their numerical resolution through standard Newton methods quite difficult. Their formulation as conic programs alleviates this difficulty since large-scale conic optimization problems can now be solved in a very robust and efficient manner, thanks to the development of dedicated interior-point algorithms. In this contribution, we review old and novel formulations of various non-smooth mechanics problems including associated plasticity with nonlinear hardening, nonlinear membranes, minimal crack surfaces, and viscoplastic fluid flows.

关键词： conic programming Variational problems Non-smooth mechanics Plasticity

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SECOND-ORDER VARIATIONAL ANALYSIS IN conic programming WITH APPLICATIONS TO OPTIMALITY AND STABILITY

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SIAM JOURNAL ON OPTIMIZATION 2015年第1期25卷 76-101页

作者： Mordukhovich, Boris S. Outrata, Jiri V. Hector Ramirez, C. Wayne State Univ Dept Math Detroit MI 48202 USA Acad Sci Czech Republ Inst Informat Theory & Automat CR-18208 Prague Czech Republic Fed Univ Ctr Informat & Appl Optimizat Ballarat Vic 3353 Australia Univ Chile Dept Ingn Matemat Santiago 3 Chile Univ Chile Ctr Modelamiento Matemat CNRS UMI 2807 Santiago 3 Chile

This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a secondorder generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related to conic programming. They include verifiable conditions for isolated calmness of the considered solution maps, sharp necessary optimality conditions for a class of mathematical programs with equilibrium constraints, and characterizations of tilt-stable local minimizers for coneconstrained problems. The main results obtained in the general conic programming setting are specified for and illustrated by the second-order cone programming.

关键词： variational analysis second-order theory conic programming generalized differentiation optimality conditions isolated calmness tilt stability

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On upper bounding Shannon capacity of graph through generalized conic programming

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OPTIMIZATION LETTERS 2019年第6期13卷 1313-1323页

作者： Bi, Yingjie Tang, Ao Cornell Univ Sch Elect & Comp Engn Ithaca NY 14850 USA

The Shannon capacity of a graph is an important graph invariant in information theory that is extremely difficult to compute. The Lovasz number, which is based on semidefinite programming relaxation, is a well-known upper bound for the Shannon capacity. To improve this upper bound, previous research tried to generalize the Lovasz number using the ideas from the sum-of-squares optimization. In this paper, we consider the possibility of developing general conic programming upper bounds for the Shannon capacity, which include the previous attempts as special cases, and show that it is impossible to find better upper bounds for the Shannon capacity along this way.

关键词： Shannon capacity of graph conic programming Sum-of-squares optimization

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