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Enhanced linear reformulation for engineering optimization models with discrete and bounded continuous variables

APPLIED MATHEMATICAL MODELLING 2018年 58卷 140-157页

作者： An, Qi Fang, Shu-Cherng Li, Han -Lin Nie, Tiantian North Carolina State Univ Dept Ind & Syst Engn Raleigh NC 27695 USA Natl Chiao Tung Univ Inst Informat Management Hsinchu Taiwan Univ North Carolina Charlotte Dept Syst Engn & Engn Management Charlotte NC 28223 USA

In this paper, we significantly extend the applicability of state-of-the-art ELDP (equations for linearizing discrete product terms) method by providing a new linearization to handle more complicated non-linear terms involving both of discrete and bounded continuous variables. A general class of "representable programming problems" is formally proposed for a much wider range of engineering applications. Moreover, by exploiting the logarithmic feature embedded in the discrete structure, we present an enhanced linear reformulation model which requires half an order fewer equations than the original ELDP. Computational experiments on various engineering design problems support the superior computational efficiency of the proposed linearization reformulation in solving engineering optimization problems with discrete and bounded continuous variables. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Nonlinear discrete optimization Linear reformulation polynomial programming Signomial programming

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The sparsest solutions to Z-tensor complementarity problems

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OPTIMIZATION LETTERS 2017年第3期11卷 471-482页

作者： Luo, Ziyan Qi, Liqun Xiu, Naihua Beijing Jiaotong Univ State Key Lab Rail Traff Control & Safety Beijing 100044 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Beijing Jiaotong Univ Sch Sci Dept Math Beijing Peoples R China

Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved norm. In this paper, a special type of tensor complementarity problems with Z-tensors has been considered. Under some mild conditions, we show that to pursuit the sparsest solutions is equivalent to solving polynomial programming with a linear objective function. The involved conditions guarantee the desired exact relaxation and also allow to achieve a global optimal solution to the relaxed nonconvex polynomial programming problem. Particularly, in comparison to existing exact relaxation conditions, such as RIP-type ones, our proposed conditions are easy to verify.

关键词： Z-tensor Tensor complementarity problem Sparse solution Exact relaxation polynomial programming

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Alternative LP and SOCP Hierarchies for ACOPF Problems

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IEEE TRANSACTIONS ON POWER SYSTEMS 2017年第4期32卷 2828-2836页

作者： Kuang, Xiaolong Ghaddar, Bissan Naoum-Sawaya, Joe Zuluaga, Luis F. Lehigh Univ Dept Ind & Syst Engn Bethlehem PA 18015 USA Univ Waterloo Dept Management Sci Waterloo ON N2L 3G1 Canada Univ Western Ontario Ivey Business Sch London ON N6A 3K7 Canada

The alternating current optimal power flow (ACOPF) problem optimizes the generation and the distribution of electric energy taking into account the active and the reactive power generation limits, demand requirements, bus voltage limits, and network flowlimits. TheACOPF problem can be formulated as a nonconvex polynomial program that is generally difficult to solve due to the nonlinear power flow constraints. A recently proposed approach to globally solve the ACOPF problem is through the formulation of a hierarchy of semidefinite programs that are computationally challenging to solve for large-scale problems. In this paper, we explore a solution approach that alleviates this computational burden by using hierarchies of linear and second order cone programs and by exploiting the network structure of the transmission grid. Furthermore, we showthat the first level of the second order cone hierarchy is equivalent to solving the conic dual of the approximation that was recently proposed in the literature, which provides the optimal solution of the ACOPF problem for special network topologies.

关键词： AC optimal power flow polynomial programming second-order cone programming

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Global optimization of general nonconvex problems with intermediate polynomial substructures

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JOURNAL OF GLOBAL OPTIMIZATION 2014年第2-3期59卷 673-693页

作者： Zorn, Keith Sahinidis, Nikolaos V. Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

This work considers the global optimization of general nonconvex nonlinear and mixed-integer nonlinear programming problems with underlying polynomial substructures. We incorporate linear cutting planes inspired by reformulation-linearization techniques to produce tight subproblem formulations that exploit these underlying structures. These cutting plane strategies simultaneously convexify linear and nonlinear terms from multiple constraints and are highly effective at tightening standard linear programming relaxations generated by sequential factorable programming techniques. Because the number of available cutting planes increases exponentially with the number of variables, we implement cut filtering and selection strategies to prevent an exponential increase in relaxation size. We introduce algorithms for polynomial substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in BARON at every node in the branch-and-bound tree. A computational study including randomly generated problems of varying size and complexity demonstrates that the exploitation of underlying polynomial substructures significantly reduces computational time, branch-and-bound tree size, and required memory.

关键词： Branch-and-bound global optimization Factorable polyhedral relaxation Reformulation-linearization techniques polynomial programming

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Exploiting equalities in polynomial programming

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OPERATIONS RESEARCH LETTERS 2008年第2期36卷 223-228页

作者： Pena, Javier F. Vera, Juan C. Zuluaga, Luis F. Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15232 USA Carnegie Mellon Univ Mellon Coll Sci Pittsburgh PA 15213 USA Univ New Brunswick Fac Business Adm Fredericton NB E3B 5A3 Canada

We propose a novel approach for solving polynomial programs over compact domains with equality constraints. By means of a genetic transformation, we show that existing solution schemes for the, typically simpler, problem without equalities can be used to address the problem with equalities. (C) 2007 Elsevier B.V. All rights reserved.

关键词： polynomial programming 0-1 programming stable set problem LMI approximations

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Processor Frequency Assignment in Three-Dimensional MPSoCs under Thermal Constraints by polynomial programming

Processor Frequency Assignment in Three-Dimensional MPSoCs u...

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IEEE Asia Pacific Conference on Circuits and Systems (APCCAS 2008)

作者： Zhao, Guangyao Kwan, Hing-Kit Lei, Chi-Un Wong, Ngai Univ Hong Kong Dept Elect & Elect Engn Hong Kong Hong Kong Peoples R China

ISBN: (纸本)9781424423415

The operating frequency and the number of cores and active layers in Multi-Processor Systems-on-Chips (MPSoC) continue to increase, resulting in rising power density and operating temperature on the die. The increasing temperature reduces the system reliability and performance and increases the cooling cost. Most traditional MPSoC thermal optimization tools are based on two-dimensional planar IC thermal modeling, which is insufficient to capture the different thermal characteristics of three-dimensional(3D) MPSoCs and the behavior subjected to Dynamic Voltage and Frequency Scaling(DVFS). The recently proposed 3D optimization approach still ignores the mutual thermal impact between cores in the same layer, which reduces the precision of frequency assignment and may cause violation in the maximum temperature constraint. In this paper, we propose an approach based on polynomial programming that addresses the problem of processor frequency assignment in 3D MPSoC system, such that the total system performance can be maximized as well as the temperature and power constraints are met for all time instances. We also compare the performance improvement of DVFS in 3D MPSoC with that in 2D MPSoC, studying the importance of intra-layer temperature correlation in 3D MPSoC.

关键词： Multi-Processor Systems-on-Chips 3D thermal model polynomial programming frequency assignment

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SECOND-ORDER CONE RELAXATIONS FOR BINARY QUADRATIC polynomial PROGRAMS

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SIAM JOURNAL ON OPTIMIZATION 2011年第1期21卷 391-414页

作者： Ghaddar, Bissan Vera, Juan C. Anjos, Miguel F. Univ Waterloo Dept Management Sci Waterloo ON N2L 3G1 Canada Tilburg Univ Tilburg Sch Econ & Management NL-5000 LE Tilburg Netherlands Ecole Polytech Montreal Dept Math & Genie Ind Montreal PQ H3C 3A7 Canada Ecole Polytech Montreal Gerad Montreal PQ H3C 3A7 Canada

Several types of relaxations for binary quadratic polynomial programs can be obtained using linear, second-order cone, or semidefinite techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our framework, we re-derive previous relaxation schemes and provide new ones. In particular, we present three relaxations for binary quadratic polynomial programs. The first two relaxations, based on second-order cone and semidefinite programming, represent a significant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems. From a practical point of view, due to the computational cost, semidefinite-based relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances. To improve the computational efficiency for solving such problems, we propose a third relaxation based purely on second-order cone programming. Computational tests on different classes of nonconvex binary quadratic polynomial problems, including quadratic knapsack problems, show that the second-order-cone-based relaxation outperforms the semidefinite-based relaxations that are proposed in the literature in terms of computational efficiency, and it is comparable in terms of bounds.

关键词： binary quadratic polynomial program polynomial programming sum-of-squares second-order cone

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Robust stability and instability of biochemical networks with parametric uncertainty

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AUTOMATICA 2011年第6期47卷 1139-1146页

作者： Waldherr, Steffen Allgoewer, Frank Univ Stuttgart Inst Syst Theory & Automat Control Stuttgart Germany

Parameter perturbations in dynamical models of biochemical networks affect the qualitative dynamical behaviour observed in the model. Since this qualitative behaviour is in many cases the key model output used to explain biological function, the robustness analysis of the model's behaviour with respect to parametric uncertainty is a crucial step in systems biology research. In this paper, we develop a new method for robustness analysis of the dynamical behaviour. As a first step, we provide a characterization of non-robust perturbations as a system of polynomial equalities and inequalities. In the second step, we apply the Positivstellensatz and Handelman representation of polynomials to check for the non-existence of solutions to this system, which can be relaxed to solving a linear program. Thereby, a solution to the linear program yields a robustness certificate for the considered dynamical behaviour. With these robustness certificates, we propose an algorithm to compute a lower robustness bound corresponding to a level of parametric uncertainty up to which no local bifurcations can occur. The applicability of the proposed method to biochemical network models is illustrated by analysing the robustness of oscillations in a model of the NF-kappa B signalling pathway. The results may be used to define a level of confidence in the observed model behaviour under parametric uncertainty, making them valuable for evaluating dynamical models of biological networks. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Analysis of systems with uncertainties Biochemical network dynamics Bifurcation analysis polynomial programming

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An optimization approach to weak approximation of stochastic differential equations with jumps

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APPLIED NUMERICAL MATHEMATICS 2011年第5期61卷 641-650页

作者： Kashima, Kenji Kawai, Reiichiro Univ Leicester Dept Math Leicester LE1 7RH Leics England Tokyo Inst Technol Grad Sch Informat Sci & Engn Tokyo 1528552 Japan

We propose an optimization approach to weak approximation of stochastic differential equations with jumps. A mathematical programming technique is employed to obtain numerically upper and lower bound estimates of the expectation of interest, where the optimization procedure ends up with a polynomial programming. A major advantage of our approach is that we do not need to simulate sample paths of jump processes, for which few practical simulation techniques exist. We provide numerical results of moment estimations for Doleans-Dade stochastic exponential, truncated stable Levy processes and Ornstein-Uhlenbeck-type processes to illustrate that our method is able to capture very well the distributional characteristics of stochastic differential equations with jumps. (C) 2011 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Doleans-Dade stochastic exponential Levy processes Stochastic differential equations Truncated stable process Ornstein-Uhlenbeck-type process polynomial programming Weak approximation

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GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2003年第2期29卷 165-194页

作者： Henrion, D Lasserre, JB CNRS LAAS F-31077 Toulouse France Acad Sci Czech Republ Inst Informat Theory & Automat CR-18208 Prague Czech Republic

GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality relaxations of the ( generally nonconvex) global optimization problem of minimizing a multivariable polynomial function subject to polynomial inequality, equality, or integer constraints. It generates a series of lower bounds monotonically converging to the global optimum without any problem splitting. Global optimality is detected and isolated optimal solutions are extracted automatically. Numerical experiments show that for most of the small-scale problems described in the literature, the global optimum is reached at low computational cost.

关键词： algorithms documentation experimentation polynomial programming semidefinite programming linear matrix inequality Matlab SeDuMi

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