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On the solution existence to convex polynomial programs and its applications

OPTIMIZATION LETTERS 2021年第2期15卷 719-731页

作者： Nguyen, Nang Tam Tran, Van Nghi Duy Tan Univ Inst Theoret & Appl Res Hanoi 100000 Vietnam Duy Tan Univ Fac Nat Sci Da Nang 550000 Vietnam Hanoi Pedag Univ 2 Hanoi Vietnam

In this paper, we present necessary/sufficient conditions for the convex polynomial programming (CPP) problems. Some new stability results for parametric CPP problems are characterized under a regular condition. We give a positive answer for the open question in Kim et al. (Optim Lett 6:363-373, 2012) for the solution existence of convex quadratic programming problems.

关键词： convex polynomial programming Solution existence Eaves type theorem Stability convex quadratic programming

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Higher-degree stochastic dominance optimality and efficiency

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2017年第3期261卷 984-993页

作者： Fang, Yi Post, Thierry Jilin Univ Ctr Quantitat Econ Changchun 130012 Peoples R China Jilin Univ Business Sch Changchun 130012 Peoples R China Nazarbayev Univ Grad Sch Business Finance Astana 010000 Kazakhstan

We characterize a range of Stochastic Dominance (SD) relations by means of finite systems of convex inequalities. For 'SD optimality' of degree 1 to 4 and 'SD efficiency' of degree 2 to 5, we obtain exact systems that can be implemented using Linear programming or convex quadratic programming. For SD optimality of degree five and higher, and SD efficiency of degree six and higher, we obtain necessary conditions. We use separate model variables for the values of the derivatives of all relevant orders at all relevant outcome levels, which allows for preference restrictions beyond the standard sign restrictions. Our systems of inequalities can be interpreted in terms of piecewise polynomial utility functions with a number of pieces that increases with the number of outcomes and the degree of SD. An empirical study analyzes the relevance of higher-order risk preferences for comparing a passive stock market index with actively managed stock portfolios in standard data sets from the empirical asset pricing literature. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Decision analysis Stochastic dominance Expected utility Linear programming convex quadratic programming

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PAL-Hom method for QP and an application to LP

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2019年第1期73卷 311-352页

作者： Wang, Guoqiang Yu, Bo Dalian Univ Technol Sch Math Sci Dalian 116024 Liaoning Peoples R China

In this paper, a proximal augmented Lagrangian homotopy (PAL-Hom) method for solving convex quadratic programming problems is proposed. This method takes the proximal augmented Lagrangian method as the outer iteration. To solve the proximal augmented Lagrangian subproblems, a homotopy method is presented as the inner iteration. The homotopy method tracks the piecewise-linear solution path of a parametric quadratic programming problem whose start problem takes an approximate solution as its solution and the target problem is the subproblem to be solved. To improve the performance of the homotopy method, the accelerated proximal gradient method is used to obtain a fairly good approximate solution that implies a good prediction of the optimal active set. Moreover, a sorting technique for the Cholesky factor update as well as an epsilon-relaxation technique for checking primal-dual feasibility and correcting the active sets are presented to improve the efficiency and robustness of the homotopy method. Simultaneously, a proximal-point-based AL-Hom method which is shown to converge in finite number of steps, is applied to linear programming. Numerical experiments on randomly generated problems and the problems from the CUTEr and Netlib test collections, support vector machines (SVMs) and contact problems of elasticity demonstrate that PAL-Hom is faster than the active-set methods and the parametric active set methods and is competitive to the interior-point methods and the specialized algorithms designed for specific models (e.g., sequential minimal optimization method for SVMs).

关键词： convex quadratic programming Linear programming Proximal point method Augmented Lagrangian method Homotopy

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Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization

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OPTIMIZATION METHODS & SOFTWARE 1999年第1-4期11-2卷 275-302页

作者： Altman, A Gondzio, J Polish Acad Sci Syst Res Inst PL-01447 Warsaw Poland Univ Edinburgh Dept Math & Stat Edinburgh EH9 3JZ Midlothian Scotland

This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidefinite systems known to be strongly factorizable to a form of Cholesky-like factorization. Two different regularization techniques, primal and dual, are very well suited to the (infeasible) primal-dual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs.

关键词： linear programming convex quadratic programming symmetric quasidefinite systems primal-dual regularization primal-dual interior point method multiple centrality correctors

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AN O(N(3)L) PRIMAL DUAL POTENTIAL REDUCTION ALGORITHM FOR SOLVING convex quadratic PROGRAMS

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MATHEMATICAL programming 1993年第2期61卷 161-170页

作者： GOLDFARB, D LIU, SC BARRA INT INC BERKELEYCA

In this paper we combine partial updating and an adaptation of Anstreicher's safeguarded linesearch of the primal-dual potential function with Kojima, Mizuno and Yoshise's potential reduction algorithm for the linear complementarity problem to obtain an O(n3 L) algorithm for convex quadratic programming. Our modified algorithm is a long step method that requires at most O(sqaure-root n L) steps.

关键词： INTERIOR POINT METHOD KARMARKAR METHOD PRIMAL DUAL POTENTIAL FUNCTION convex quadratic programming RANK-ONE UPDATE

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On convex quadratic programs with linear complementarity constraints

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2013年第3期54卷 517-554页

作者： Bai, Lijie Mitchell, John E. Pang, Jong-Shi Rensselaer Polytech Inst Dept Math Sci Troy NY 12180 USA Univ Illinois Dept Ind & Enterprise Syst Engn Urbana IL 61801 USA

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one for each complementarity constraint. Based on the primal-dual relationship of a pair of convex quadratic programs and on a logical Benders scheme, an extreme ray/point generation procedure is developed, which relies on valid satisfiability constraints for the integer program. To improve this scheme, we propose a two-stage approach wherein the first stage solves the mixed integer quadratic program with pre-set upper bounds on the complementarity variables, and the second stage solves the program outside this bounded region by the Benders scheme. We report computational results with our method. We also investigate the addition of a penalty term y (T) Dw to the objective function, where y and w are the complementary variables and D is a nonnegative diagonal matrix. The matrix D can be chosen effectively by solving a semidefinite program, ensuring that the objective function remains convex. The addition of the penalty term can often reduce the overall runtime by at least 50 %. We report preliminary computational testing on a QP relaxation method which can be used to obtain better lower bounds from infeasible points;this method could be incorporated into a branching scheme. By combining the penalty method and the QP relaxation method, more than 90 % of the gap can be closed for some QPCC problems.

关键词： convex quadratic programming Logical Benders decomposition Satisfiability constraints Semidefinite programming

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Non-negative mixed finite element formulations for a tensorial diffusion equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2009年第18期228卷 6726-6752页

作者： Nakshatrala, K. B. Valocchi, A. J. Texas A&M Univ Dept Mech Engn College Stn TX 77843 USA Univ Illinois Dept Civil & Environm Engn Urbana IL 61801 USA

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart-Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum-minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. In this paper, we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems. Published by Elsevier Inc.

关键词： Maximum-minimum principles for elliptic PDEs Discrete maximum-minimum principle Non-negative solutions Active set strategy convex quadratic programming Tensorial diffusion equation Monotone methods

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An outcome-space finite algorithm for solving linear multiplicative programming

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期179卷 494-505页

作者： Gao, Yuelin Xu, Chengxian Yang, Yongjian Xi An Jiao Tong Univ Sch Finance & Econ Xian 710049 Peoples R China NW Secondly Natl Coll Dept Informat & Computat Sci Yin Chuan 750021 Peoples R China Shanghai Univ Dept Math Shanghai 200436 Peoples R China

This paper presents an outcome-space finite algorithm for solving linear multiplicative programming, in each iteration of which a convex quadratic programming is only solved. In the paper, we give a global optimization condition on a class of multiplicative programming problems and prove that the proposed algorithm is finite terminative and gain a global optimal solution of the former problem when it stops. It can be shown by the numerical results that the proposed algorithm is effective and the computational results can be gained in short time. (c) 2005 Elsevier Inc. All rights reserved.

关键词： linear multiplicative programming convex quadratic programming outcome-space global optimization outer approximation method

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Componentwise error bounds for linear complementarity problems

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IMA JOURNAL OF NUMERICAL ANALYSIS 2011年第1期31卷 348-357页

作者： Wang, Zhengyu Yuan, Ya-Xiang Nanjing Univ Dept Math Nanjing 210093 Peoples R China Univ Karlsruhe Karlsruhe Inst Technol Inst Appl & Numer Math D-76128 Karlsruhe Germany Chinese Acad Sci Acad Math & Syst Sci Lab Sci & Engn Comp Inst Computat Math & Sci Engn Comp Beijing 100190 Peoples R China

Componentwise error bounds for linear complementarity problems are presented. For the problem with an H-matrix the error bound can be computed by solving a system of linear equations. It is proved that our error bound is more accurate than that obtained recently by Chen & Xiang (2006, Math. Prog., Ser. A, 106, 513-525). Numerical results show that the new bound is often much better than previous ones.

关键词： error bound linear complementarity problem convex quadratic programming

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Proximal Stabilized Interior Point Methods and Low-Frequency-Update Preconditioning Techniques

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第3期197卷 1061-1103页

作者： Cipolla, Stefano Gondzio, Jacek Univ Edinburgh Sch Math Edinburgh Scotland

In this work, in the context of Linear and convex quadratic programming, we consider Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is strongly supported by theoretical results concerning convergence and the rate of convergence, and can handle degenerate problems. Moreover, in the second part of this work, we analyse the interactions between the regularization parameters and the computational footprint of the linear algebra routines used to solve the Newton linear systems. In particular, when these systems are solved using an iterative Krylov method, we are able to show-using a new rearrangement of the Schur complement which exploits regularization-that general purposes preconditioners remain attractive for a series of subsequent IPM iterations. Indeed, if on the one hand a series of theoretical results underpin the fact that the approach here presented allows a better re-use of such computed preconditioners, on the other, we show experimentally that such (re)computations are needed only in a fraction of the total IPM iterations. The resulting regularized second order methods, for which low-frequency-update of the preconditioners are allowed, pave the path for an alternative class of second order methods characterized by reduced computational effort.

关键词： Interior point methods Proximal point methods Regularized primal-dual methods convex quadratic programming

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