Global Optimization for Generalized Geometric Programs with Mixed Free-Sign Variables

Li, Hanlin; Lu, Hao-Chun

doi:10.1287/opre.1080.0586

Cited by 39 publications

(14 citation statements)

References 17 publications

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“…The basis for our analysis will be a reformulation of D n which has been part of the mathematical programming folklore since at least 1972 [13] and which has recently been re-discovered by several authors [5,15,17]. This formulation requires a set of vectors…”

Section: Encoding Formulations and Branch-and-boundmentioning

confidence: 99%

“…This choice transforms (2) into a formulation with a logarithmic number of binary variables which was the original case studied in [5,13,15,17]. The induced constraint branching scheme is not quite clear, but we can easily see that it creates balanced branch-and-bound trees by observing that we have…”

Section: Encoding Formulations and Branch-and-boundmentioning

confidence: 99%

“…These new variables are sometimes called by-variables and lead to balanced branch-and-bound trees that can significantly reduce solution times [6][7][8][9][10][11][12]. The third approach proposes to model n alternatives with a number of binary variables that is logarithmic in n [5, [13][14][15][16][17]. This approach tends to generate more balanced branch-and-bound trees, but, as we will see in Section 2, they are not immune to imbalance issues either.…”

Section: Introductionmentioning

confidence: 99%

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Incremental and encoding formulations for Mixed Integer Programming

Yıldız

Vielma

2013

Operations Research Letters

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The standard way to represent a choice between n alternatives in Mixed Integer Programming is through n binary variables that add up to one. Unfortunately, this approach commonly leads to unbalanced branch-and-bound trees and diminished solver performance. In this paper, we present an encoding formulation framework that encompasses and expands existing approaches to mitigate this behavior. Through this framework, we generalize the incremental formulation for piecewise linear functions to any finite union of polyhedra with identical recession cones.

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Section: Encoding Formulations and Branch-and-boundmentioning

confidence: 99%

Section: Encoding Formulations and Branch-and-boundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Incremental and encoding formulations for Mixed Integer Programming

Yıldız

Vielma

2013

Operations Research Letters

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“…A similar technique is used in [105,103,41] to model the constraint programming all-different requirement [75] in problems such as graph coloring. For more general problems, MIP formulations with a logarithmic number of binary variables were originally considered in [81,148] and have received significant attention recently [155,127,156,107,74,6,125,159,160].…”

Section: Incremental Formulationsmentioning

confidence: 99%

“…Recently, formulations that are essentially identical to this one were independently proposed in [6,74,107,159,160]. However, the basic idea behind formulation (9.3) has in fact been part of the mathematical programming folklore for a long time.…”

Section: Incremental Formulationsmentioning

confidence: 99%

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

268

143

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1. Introduction. Throughout more than 50 years of existence, mixed integer linear programming (MIP) theory and practice have been significantly developed, and it is now an indispensable tool in business and engineering [68,94,104]. Two reasons for the success of MIP are linear programming (LP) based solvers and the modeling flexibility of MIP. We now have several extremely effective state-of-the-art solvers [82,69, 52,171] that incorporate many advanced techniques [1,2,25,23,92,112,24] and, since its early stages, MIP has been used to model a wide range of applications [44,45].While in many cases constructing valid MIP formulations is relatively straightforward, some care should be taken in this construction as certain formulation attributes can significantly reduce the effectiveness of LP-based solvers. Fortunately, constructing formulations that behave well with state-of-the-art solvers can usually be achieved by following simple guidelines described in standard textbooks. However, more advanced techniques can often perform significantly better than textbook formulations and are sometimes a necessity. The main objective of this survey is to summarize the state of the art of such formulation techniques for a wide range of problems. To keep the length of this survey under control, we concentrate on formulations for sets of a mixed integer nature that require both integer constrained and continuous variables. We hence purposefully place less emphasis on some related areas such as combinatorial optimization, quadratic and polynomial 0/1 optimization, and polyhedral approximations of convex sets. These topics are certainly areas of important and active research, so we cover them succinctly in section 12.

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Control of tree water networks: A geometric programming approach

Perelman

Amin

2015

Water Resources Research

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This paper presents a modeling and operation approach for tree water supply systems. The network control problem is approximated as a geometric programming (GP) problem. The original nonlinear nonconvex network control problem is transformed into a convex optimization problem. The optimization model can be efficiently solved to optimality using state-of-the-art solvers. Two control schemes are presented: (1) operation of network actuators (pumps and valves) and (2) controlled demand shedding allocation between network consumers with limited resources. The dual of the network control problem is formulated and is used to perform sensitivity analysis with respect to hydraulic constraints. The approach is demonstrated on a small branched-topology network and later extended to a medium-size irrigation network. The results demonstrate an intrinsic trade-off between energy costs and demand shedding policy, providing an efficient decision support tool for active management of water systems.

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Global Optimization for Generalized Geometric Programs with Mixed Free-Sign Variables

Cited by 39 publications

References 17 publications

Incremental and encoding formulations for Mixed Integer Programming

Incremental and encoding formulations for Mixed Integer Programming

Mixed Integer Linear Programming Formulation Techniques

Control of tree water networks: A geometric programming approach

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