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On solving a non-convex quadratic programming problem involving resistance distances in graphs

ANNALS OF OPERATIONS RESEARCH 2020年第2期287卷 643-651页

作者： Dubey, Dipti Neogy, S. K. Indian Stat Inst 7 SJS Sansanwal Marg New Delhi 110016 India

quadratic programming problems involving distance matrix (D) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12-20, 2012), Bapat and Neogy (Ann Oper Res 243:365-373, 2016). In this paper, we consider the question of solving the quadratic programming problem of finding maximum of x(T)Rx subject to x being a nonnegative vector with sum 1 and show that for the class of simple graphs with resistance distance matrix (R) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratic programming problem. An application to symmetric bimatrix game is also presented.

关键词： Resistance distance Laplacian matrix non-convex quadratic programming Polynomial time algorithm Symmetric bimatrix game

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An Optimization Setup of the Decarbonization Problem in the Transportation Sector 11th

An Optimization Setup of the Decarbonization Problem in the ...

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11th International Conference on Network Games, Control and Optimization

作者： Couthures, Anthony Vogler, Prunelle Beaude, Olivier Lasaulce, Samson Morarescu, Irinel-Constantin Univ Lorraine CNRS CRAN F-54000 Nancy France EDF Lab Paris Saclay OSIRIS Dept Palaiseau France Khalifa Univ Abu Dhabi U Arab Emirates Tech Univ Cluj Napoca Automat Dept Cluj Napoca Romania

ISBN: (纸本)9783031785993;9783031786006

This paper proposes an optimization-based formulation of the decarbonization problem for the transportation sector, one of the main sources of CO2 emissions. An important feature of the proposed approach is that a detailed model is considered i.e., with different transportation modes and compositions of the associated fleets of vehicles. Using real data, this optimization problem is solved numerically under monetary and CO2 emissions constraints, but also constraints ensuring the feasibility of the transition. The obtained control actions provide insights into the transition from existing transportation modes to more sustainable ones. A discussion is made on the complexity of this problem: the dependency of resolution time on the budget values is in particular numerically assessed.

关键词： Decarbonization non-convex quadratic programming Resource allocation

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A new algorithm for concave quadratic programming

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第3期75卷 655-681页

作者： Zamani, Moslem Ton Duc Thang Univ Parametr MultiObject Optimizat Res Grp Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Math & Stat Ho Chi Minh City Vietnam

The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a modified version by restricting the feasible set. This leads to a new bound for quadratic programs. We demonstrate that the dual of the bound is a semi-definite relaxation of quadratic programs. In addition, we probe the relationship between this bound and the well-known bounds in the literature. In the second part, thanks to the new bound, we propose a branch and cut algorithm for concave quadratic programs. We establish that the algorithm enjoys global convergence. The effectiveness of the method is illustrated for numerical problem instances.

关键词： non-convex quadratic programming Duality Semi-definite relaxation Bound Branch and cut method Concave quadratic programming

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Globally Solving nonconvex quadratic Programs via Linear Integer programming Techniques

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INFORMS JOURNAL ON COMPUTING 2020年第1期32卷 40-56页

作者： Xia, Wei Vera, Juan Zuluaga, Luis F. Lehigh Univ Ind & Syst Engn Dept Bethlehem PA 18015 USA Tilburg Univ Tilburg Sch Econ & Management Econometr & Operat NL-5037 AB Tilburg Netherlands

We reformulate a (indefinite) quadratic program (QP) as a mixed-integer linear programming (MILP) problem by first reformulating a QP as a linear complementary problem, and then using binary variables and big-M constraints to model its complementary constraints. To obtain such reformulation, we use fundamental results on the solution of perturbed linear systems to impose bounds on the QP's dual variables without eliminating any of its (globally) optimal primal solutions. Reformulating a nonconvex QP as a MILP problem allows the use of current state-of-the-art MILP solvers to find its global optimal solution. To illustrate this, we compare the performance of this MILP-based solution approach, labeled quadprogIP, with quadprogBB, BARON, and CPLEX. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. Also, unlike quadprogBB, quadprogIP is able to solve QP instances in which the dual feasible set is unbounded. The MATLAB code quadprogIP and the instances used to perform the reported numerical experiments are publicly available at https://***/xiawei918/quadprogIP.

关键词： non-convex quadratic programming global optimization mixed-integer linear programming KKT conditions branch and bound Hoffman bound

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Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint

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JOURNAL OF GLOBAL OPTIMIZATION 2012年第2期54卷 275-293页

作者： Feng, Joe-Mei Lin, Gang-Xuan Sheu, Reuy-Lin Xia, Yong Natl Cheng Kung Univ Dept Math Tainan 70101 Taiwan Beihang Univ Sch Math & Syst Sci Minist Educ LMIB Beijing Peoples R China

This paper extends and completes the discussion by Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted) about the quadratic programming over one quadratic constraint (QP1QC). In particular, we relax the assumption to cover more general cases when the two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence. Under such an assumption, the nonconvex (QP1QC) problem can be solved through a dual approach with no duality gap. This is unusual for general nonconvex programming but we can explain by showing that (QP1QC) is indeed equivalent to a linearly constrained convex problem, which happens to be dual of the dual of itself. Another type of hidden convexity can be also found in the boundarification technique developed in Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted).

关键词： non-convex quadratic programming Simultaneously diagonalizable via congruence Slater's condition Duality Hidden convexity

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Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint

Duality and solutions for quadratic programming over single ...

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1st World Congress on Global Optimization in Engineering & Science (WCGO)

作者： Feng, Joe-Mei Lin, Gang-Xuan Sheu, Reuy-Lin Xia, Yong Natl Cheng Kung Univ Dept Math Tainan 70101 Taiwan Beihang Univ Sch Math & Syst Sci Minist Educ LMIB Beijing Peoples R China

关键词： non-convex quadratic programming Simultaneously diagonalizable via congruence Slater's condition Duality Hidden convexity

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A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2018年第1期70卷 33-59页

作者： Xu, Guanglin Burer, Samuel Univ Iowa Dept Management Sci Iowa City IA 52242 USA

We study two-stage adjustable robust linear programming in which the right-hand sides are uncertain and belong to a convex, compact uncertainty set. This problem is NP-hard, and the affine policy is a popular, tractable approximation. We prove that under standard and simple conditions, the two-stage problem can be reformulated as a copositive optimization problem, which in turn leads to a class of tractable, semidefinite-based approximations that are at least as strong as the affine policy. We investigate several examples from the literature demonstrating that our tractable approximations significantly improve the affine policy. In particular, our approach solves exactly in polynomial time a class of instances of increasing size for which the affine policy admits an arbitrarily large gap.

关键词： Two-stage adjustable robust optimization Robust optimization Bilinear programming non-convex quadratic programming Semidefinite programming Copositive programming

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Robust sensitivity analysis of the optimal value of linear programming

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OPTIMIZATION METHODS & SOFTWARE 2017年第6期32卷 1187-1205页

作者： Xu, Guanglin Burer, Samuel Univ Iowa Dept Management Sci Iowa City IA 52242 USA

We propose a framework for sensitivity analysis (SA) of linear programs (LPs) in minimization form, allowing for simultaneous perturbations in the objective coefficients and right-hand sides, where the perturbations are modelled in a compact, convex, and tractable uncertainty set. This framework unifies and extends multiple approaches for LP SA in the literature and has close ties to worst-case linear optimization and two-stage adaptive optimization. We define the minimum (best-case) and maximum (worst-case) LP optimal values, p-and p(+), over the uncertainty set, and we discuss issues of finiteness, attainment, and computational complexity. While p-and p(+) are difficult to compute in general, we prove that they equal the optimal values of two separate, but related, copositive programs. We then develop tight, tractable conic relaxations to provide lower and upper bounds on p-and p(+), respectively. We also develop techniques to assess the quality of the bounds, and we validate our approach computationally on several examples from-and inspired by-the literature. We find that the bounds on p-and p(+) are very strong in practice and, in particular, are at least as strong as known results for specific cases from the literature.

关键词： sensitivity analysis minimax problem non-convex quadratic programming semidefinite programming copositive programming uncertainty set

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Global optimality condition for quadratic optimization problems under data uncertainty

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POSITIVITY 2021年第3期25卷 1027-1044页

作者： Barro, Moussa Ouedraogo, Ali Traore, Sado Univ Nazi Boni Dept Math Bobo Dioulasso Burkina Faso Inst Sci Dept Math Ouagadougou Burkina Faso

In this paper, we establish a robust version of the S-lemma that we use to characterize robust solutions for classes of homogeneous and non-homogeneous quadratic problems with a quadratic inequality constraint under interval uncertainty and a linear equality constraint. Necessary and sufficient conditions of global optimality of robust solution of these problems are given.

关键词： Robust optimization non-convex quadratic programming Data interval uncertainty Optimality conditions

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Least Squares Method for Solving Fuzzy LR Interval Algebraic Linear Systems

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FUZZY INFORMATION AND ENGINEERING 2022年第3期14卷 349-359页

作者： Salari, Mehrnoosh Ghanbari, Reza Ghorbani-Moghadam, Khatere Ferdowsi Univ Mashhad Dept Math Sci Mashhad Razavi Khorasan Iran Kharazmi Univ Mosaheb Inst Math Tehran Iran

We first investigate the solvability conditions of fuzzy LR interval algebraic linear systems with fuzzy LR interval coefficient matrix and fuzzy LR interval hand-right vector. Next, we compute the solution of FLRIALS (fuzzy LR interval algebraic linear systems) using the least squares approach. We show that solving the least squares problem is equivalent to solving a quadratic programming problem. Indeed, instead of the FLRIALS, this quadratic programming has a non-convex constraint. Thus, we relax this constraint to have a convex search space. Then, from the solution obtained from quadratic programming on convex search space, we give an approximate solution for FLRIALS.

关键词： Fuzzy LR interval number fuzzy LR interval algebraic linear system least squares method non-convex quadratic programming

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