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33
Algorithms for the Satisfiability (SAT) Problem: A Survey
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1996
"... . The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, compute ..."
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Cited by 127 (3 self)
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. The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a unified perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the sat...
Feature space interpretation of svms with indefinite kernels
 IEEE Trans Pattern Anal Mach Intell
, 2005
"... Abstract—Kernel methods are becoming increasingly popular for various kinds of machine learning tasks, the most famous being the support vector machine (SVM) for classification. The SVM is well understood when using conditionally positive definite (cpd) kernel functions. However, in practice, noncp ..."
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Cited by 57 (2 self)
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Abstract—Kernel methods are becoming increasingly popular for various kinds of machine learning tasks, the most famous being the support vector machine (SVM) for classification. The SVM is well understood when using conditionally positive definite (cpd) kernel functions. However, in practice, noncpd kernels arise and demand application in SVMs. The procedure of “plugging ” these indefinite kernels in SVMs often yields good empirical classification results. However, they are hard to interpret due to missing geometrical and theoretical understanding. In this paper, we provide a step toward the comprehension of SVM classifiers in these situations. We give a geometric interpretation of SVMs with indefinite kernel functions. We show that such SVMs are optimal hyperplane classifiers not by margin maximization, but by minimization of distances between convex hulls in pseudoEuclidean spaces. By this, we obtain a sound framework and motivation for indefinite SVMs. This interpretation is the basis for further theoretical analysis, e.g., investigating uniqueness, and for the derivation of practical guidelines like characterizing the suitability of indefinite SVMs. Index Terms—Support vector machine, indefinite kernel, pseudoEuclidean space, separation of convex hulls, pattern recognition. æ 1
A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs : II. Application of Theory and Test Problems
 Engng
, 1990
"... In Part I (Floudas and Visweswaran, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until th ..."
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Cited by 54 (21 self)
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In Part I (Floudas and Visweswaran, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until the upper and lower bounds from these problems converged to an fflglobal optimum. In this paper, theoretical results are presented for several classes of mathematical programming problems that include : (i) the general quadratic programming problem, (ii) quadratic programming problems with quadratic constraints, (iii) pooling and blending problems, and (iv) unconstrained and constrained optimization problems with polynomial terms in the objective function and/or constraints. For each class, a few examples are presented illustrating the approach. Keywords : Global Optimization, Quadratic Programming, Quadratic Constraints, Polynomial functions, Pooling and Blending Problems. Author to whom...
Quadratic Optimization
, 1995
"... . Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, t ..."
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Cited by 46 (3 self)
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. Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, the quadratic problem is known to be NPhard, which makes this one of the most interesting and challenging class of optimization problems. In this chapter, we review various properties of the quadratic problem, and discuss different techniques for solving various classes of quadratic problems. Some of the more successful algorithms for solving the special cases of bound constrained and large scale quadratic problems are considered. Examples of various applications of quadratic programming are presented. A summary of the available computational results for the algorithms to solve the various classes of problems is presented. Key words: Quadratic optimization, bilinear programming, concave pro...
A PrimalRelaxed Dual Global Optimization Approach
, 1993
"... A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, convert the original problem into primal and relaxed dual subproblems that provide ..."
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Cited by 41 (19 self)
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A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, convert the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of fflfinite convergence and fflglobal optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.
An Interior Point Algorithm to Solve Computationally Difficult Set Covering Problems
, 1990
"... ..."
A Review Of Techniques In The Verified Solution Of Constrained Global Optimization Problems
, 1996
"... Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previousl ..."
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Cited by 25 (6 self)
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Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previously developed algorithms and general work on the subject are also listed. Limitations of present knowledge are mentioned, and advice is given on which techniques to use in various contexts. Applications are discussed. 1 INTRODUCTION, BASIC IDEAS AND LITERATURE We consider the constrained global optimization problem minimize OE(X) subject to c i (X) = 0; i = 1; : : : ; m (1.1) a i j x i j b i j ; j = 1; : : : ; q; where X = (x 1 ; : : : ; xn ) T . A general constrained optimization problem, including inequality constraints g(X) 0 can be put into this form by introducing slack variables s, replacing by s + g(X) = 0, and appending the bound constraint 0 s ! 1; see x2.2. 2 Chapter 1 W...
P.: Robust control via sequential semidefinite programming
 SIAM J. Control Optim
, 2002
"... Abstract. This paper discusses nonlinear optimization techniques in robust control synthesis, with special emphasis on design problems which may be cast as minimizing a linear objective function under linear matrix inequality (LMI) constraints in tandem with nonlinear matrix equality constraints. Th ..."
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Cited by 24 (8 self)
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Abstract. This paper discusses nonlinear optimization techniques in robust control synthesis, with special emphasis on design problems which may be cast as minimizing a linear objective function under linear matrix inequality (LMI) constraints in tandem with nonlinear matrix equality constraints. The latter type of constraints renders the design numerically and algorithmically difficult. We solve the optimization problem via sequential semidefinite programming (SSDP), a technique which expands on sequential quadratic programming (SQP) known in nonlinear optimization. Global and fast local convergence properties of SSDP are similar to those of SQP, and SSDP is conveniently implemented with available semidefinite programming (SDP) solvers. Using two test examples, we compare SSDP to the augmented Lagrangian method, another classical scheme in nonlinear optimization, and to an approach using concave optimization. Key words. nonlinear programming, sequential semidefinite programming, robust gainscheduling control design, linear matrix inequalities, nonlinear matrix equalities
Lower bounds for the quadratic assignment problem
 University of Munich
, 1994
"... Abstract. We investigate the classical GilmoreLawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the GilmoreLawler Bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new ..."
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Cited by 20 (5 self)
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Abstract. We investigate the classical GilmoreLawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the GilmoreLawler Bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branchandbound type algorithm for the quadratic assignment problem. 1.
New Properties and Computational Improvement of the GOP Algorithm For Problems With Quadratic Objective Function and Constraints
 Journal of Global Optimization
, 1993
"... In Floudas and Visweswaran (1990, 1992), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series of primal and relaxed dual problems that provide valid ..."
Abstract

Cited by 20 (10 self)
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In Floudas and Visweswaran (1990, 1992), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series of primal and relaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an fflglobal optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and in...