A reformulation-convexification approach for solving nonconvex quadratic programming problems

Sherali, Hanif D.; Tuncbilek, Cihan H.

doi:10.1007/bf01100203

Cited by 147 publications

(76 citation statements)

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“…The Reformulation-Linearization Technique (RLT) of Sherali and co-workers [84,85,125,126,127,128] considers variable/variable, variable/equation, and equation/equation products that may lift the convex relaxation of MIQCQP. These products are additionally useful in the context of variable bounding [18,127,128].…”

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

“…These products are additionally useful in the context of variable bounding [18,127,128]. Because there may be many possible RLT equations in MIQCQP, multiple filtering techniques exist to automatically reduce the size of the RLT representation [124,127,128]. However, it is difficult to a priori determine which RLT cuts will be the most advantageous for a specific problem class, so there is documentation on the relative advantage of specific RLT cuts for individual problem classes [4,16,70,74,109,115,133].…”

Section: Literature Reviewmentioning

confidence: 99%

“…Variable bounding algorithms relevant to this work include: interval arithmetic bounds tightening [6,7,15,18,27,127,128], optimality-based bounds tightening [89,133], reduced cost bounds tightening [117,118], and quadratic equation constraint satisfaction [44,45,77].…”

Section: Literature Reviewmentioning

confidence: 99%

“…[125,126,127,128]. For the first n levels of the branch-and-bound tree (n defaults to 3 in the C++ implementation), GloMIQO begins by solving the basic MILP relaxation of MIQCQP.…”

Section: Adaptive Reformulation-linearization Technique (Rlt)mentioning

confidence: 99%

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GloMIQO: Global mixed-integer quadratic optimizer

2012

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This paper introduces the Global Mixed-Integer Quadratic Optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. The algorithmic components are presented for: reformulating user input, detecting special structure including convexity and edge-concavity, generating tight convex relaxations, partitioning the search space, bounding the variables, and finding good feasible solutions. To demonstrate the capacity of GloMIQO, we extensively tested its performance on a test suite of 399 problems of diverse size and structure. The test cases are taken from process networks applications, computational geometry problems, GLOBALLib, MINLPLib, and the Bonmin test set. We compare the performance of GloMIQO with respect to four state-of-the-art global optimization solvers: BARON 10.1.2,

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Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

“…[125,126,127,128]. For the first n levels of the branch-and-bound tree (n defaults to 3 in the C++ implementation), GloMIQO begins by solving the basic MILP relaxation of MIQCQP.…”

Section: Adaptive Reformulation-linearization Technique (Rlt)mentioning

confidence: 99%

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GloMIQO: Global mixed-integer quadratic optimizer

2012

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Mathematics in Chemical Engineering

Finlayson

Biegler

Grossmann

2006

Ullmann's Encyclopedia of Industrial Chemistry

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The article contains sections titled: 1. Solution of Equations 1.1. Matrix Properties 1.2. Linear Algebraic Equations 1.3. Nonlinear Algebraic Equations 1.4. Linear Difference Equations 1.5. Eigenvalues 2. Approximation and Integration 2.1. Introduction 2.2. Global Polynomial Approximation 2.3. Piecewise Approximation 2.4. Quadrature 2.5. Least Squares 2.6. Fourier Transforms of Discrete Data 2.7. Two‐Dimensional Interpolation and Quadrature 3. Complex Variables 3.1. Introduction to the Complex Plane 3.2. Elementary Functions 3.3. Analytic Functions of a Complex Variable 3.4. Integration in the Complex Plane 3.5. Other Results 4. Integral Transforms 4.1. Fourier Transforms 4.2. Laplace Transforms 4.3. Solution of Partial Differential Equations by Using Transforms 5. Vector Analysis 6. Ordinary Differential Equations as Initial Value Problems 6.1. Solution by Quadrature 6.2. Explicit Methods 6.3. Implicit Methods 6.4. Stiffness 6.5. Differential ‐ Algebraic Systems 6.6. Computer Software 6.7. Stability, Bifurcations, Limit Cycles 6.8. Sensitivity Analysis 6.9. Molecular Dynamics 7. Ordinary Differential Equations as Boundary Value Problems 7.1. Solution by Quadrature 7.2. Initial Value Methods 7.3. Finite Difference Method 7.4. Orthogonal Collocation 7.5. Orthogonal Collocation on Finite Elements 7.6. Galerkin Finite Element Method 7.7. Cubic B‐Splines 7.8. Adaptive Mesh Strategies 7.9. Comparison 7.10. Singular Problems and Infinite Domains 8. Partial Differential Equations 8.1. Classification of Equations 8.2. Hyperbolic Equations 8.3. Parabolic Equations in One Dimension 8.4. Elliptic Equations 8.5. Parabolic Equations in Two or Three Dimensions 8.6. Special Methods for Fluid Mechanics 8.7. Computer Software 9. Integral Equations 9.1. Classification 9.2. Numerical Methods for Volterra Equations of the Second Kind 9.3. Numerical Methods for Fredholm, Urysohn, and Hammerstein Equations of the Second Kind 9.4. Numerical Methods for Eigenvalue Problems 9.5. Green's Functions 9.6. Boundary Integral Equations and Boundary Element Method 10. Optimization 10.1. Introduction 10.2. Gradient Based Nonlinear Programming 10.3. Optimization Methods without Derivatives 10.4. Global Optimization 10.5. Mixed Integer Programming 10.6. Dynamic Optimization 10.7. Development of Optimization Models 11. Probability and Statistics 11.1. Concepts 11.2. Sampling and Statistical Decisions 11.3. Error Analysis in Experiments 11.4. Factorial Design of Experiments and Analysis of Variance 12. Multivariable Calculus Applied to Thermodynamics 12.1. State Functions 12.2. Applications to Thermodynamics 12.3. Partial Derivatives of All Thermodynamic Functions

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Reformulation‐Linearization Technique for MIPs

Sherali

2011

Wiley Encyclopedia of Operations Research and Management Science

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The reformulation‐linearization technique (RLT) for mixed‐integer programs is an automatic model enhancement approach that generates a hierarchy of relaxations spanning the spectrum from the continuous linear programming (LP) relaxation to the convex hull of feasible solutions. This process is applicable to both 0‐1 as well as general discrete programs, and can be further augmented through the use of classes of semidefinite cuts. Often, the first‐level relaxation itself provides a sufficiently tight model reformulation that significantly improves problem solvability. The RLT procedure also offers a unifying framework for solving continuous nonconvex factorable optimization problems to global optimality.

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A reformulation-convexification approach for solving nonconvex quadratic programming problems

Cited by 147 publications

References 28 publications

GloMIQO: Global mixed-integer quadratic optimizer

GloMIQO: Global mixed-integer quadratic optimizer

Mathematics in Chemical Engineering

Reformulation‐Linearization Technique for MIPs

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