An exact solution method for unconstrained quadratic 0–1 programming: a geometric approach

Li, D.; Sun, Xiaoling; Liu, C. L.

doi:10.1007/s10898-011-9713-2

Cited by 21 publications

(6 citation statements)

References 47 publications

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“…Ahlatçıoglu et al [1] proposed to combine QCR and the convex hull relaxation to solve problem (BQP). The geometric investigation in Li et al [36] for binary quadratic programs provides some theoretical support for QCR from another angle. Billionnet et al [9] extended the QCR approach to general mixed-integer programs by using binary decomposition.…”

Section: Combination Of Lift-and-convexification and Qcrmentioning

confidence: 98%

See 1 more Smart Citation

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Wu¹,

Sun²,

Li³

2017

SIAM J. Optim.

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We consider in this paper a class of semi-continuous quadratic programming problems which arises in many real-world applications such as production planning, portfolio selection and subset selection in regression. We propose a lift-and-convexification approach to derive an equivalent reformulation of the original problem. This liftand-convexification approach lifts the quadratic term involving x only in the original objective function f (x, y) to a quadratic function of both x and y and convexifies this equivalent objective function. While the continuous relaxation of our new reformulation attains the same tight bound as achieved by the continuous relaxation of the well known perspective reformulation, the new reformulation also retains the linearly constrained quadratic programming structure of the original mix-integer problem. This prominent feature improves the performance of branch-and-bound algorithms by providing the same tightness at the root node as the state-of-the-art perspective reformulation and offering much faster processing time at children nodes. We further combine the liftand-convexification approach and the quadratic convex reformulation approach in the literature to form an even tighter reformulation. Promising results from our computational tests in both portfolio selection and subset selection problems numerically verify the benefits from these theoretical features of our new reformulations.

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Section: Combination Of Lift-and-convexification and Qcrmentioning

confidence: 98%

“…If the objective function of (P(u, v, w, t)) is convex, by the strong duality of convex quadratic programming (see, e.g., Proposition 6.5.6 in [5]), the optimal values of (P(u, v, w, t)) and (36) are equal. Thus, we have shown that problem (MAXuvwt) is equivalent to an SDP problem in the form of (SDP a ).…”

Section: Theoremmentioning

confidence: 99%

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Wu¹,

Sun²,

Li³

2017

SIAM J. Optim.

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“…对于一般的 0-1 二次优化问题, 学者们提出了求解精确解的多种方法, 包括基于代数的方法、线性化方法、割平面算法和分支定界算法等. 文献 [370,378] [362,379,380]). [327] 使用上述半定规划松弛给出最大割问题 0.8756 的近似界.…”

Section: 非线性整数优化unclassified

Modern optimization theory and applications

Deng¹,

Jian-jun²,

Ge³

et al. 2020

Sci. Sin.-Math.

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线性规划问题可以说是目前优化领域中最重要且已经被大量研究过的一类问题. 本节将回顾 3 个线性规划相关主题的最新进展: (i) 线性规划算法; (ii) 线性规划和 Markov 决策过程; (iii) 在线线性规划. 本节主要介绍 (i) 和 (ii) 两个领域近期一些比较重要和有趣的发现. (iii) 将在下一节讨论, 希望这些讨论能激发科研工作者在这些领域进行新的研究.

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“…To solve a QUBO problem, a number of exact methods have been developed [5,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. However, due to the computational complexity of the problems, these approaches have only been able to solve smallsized instances.…”

Section: The Problem and Previous Workmentioning

confidence: 99%

Building an iterative heuristic solver for a quantum annealer

et al. 2016

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A quantum annealer heuristically minimizes quadratic unconstrained binary optimization (QUBO) problems, but is limited by the physical hardware in the size and density of the problems it can handle. We have developed a meta-heuristic solver that utilizes D-Wave Systems' quantum annealer (or any other QUBO problem optimizer) to solve larger or denser problems, by iteratively solving subproblems, while keeping the rest of the variables fixed. We present our algorithm, several variants, and the results for the optimization of standard QUBO problem instances from OR-Library of sizes 500 and 2500 as well as the Palubeckis instances of sizes 3000 to 7000. For practical use of the solver, we show the dependence of the time to best solution on the desired gap to the best known solution. In addition, we study the dependence of the gap and the time to best solution on the size of the problems solved by the underlying optimizer.Comment: 21 pages, 4 figures; minor edit

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An exact solution method for unconstrained quadratic 0–1 programming: a geometric approach

Cited by 21 publications

References 47 publications

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Modern optimization theory and applications

Building an iterative heuristic solver for a quantum annealer

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