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12
Global Optimization For Constrained Nonlinear Programming
, 2001
"... In this thesis, we develop constrained simulated annealing (CSA), a global optimization algorithm that asymptotically converges to constrained global minima (CGM dn ) with probability one, for solving discrete constrained nonlinear programming problems (NLPs). The algorithm is based on the necessary ..."
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Cited by 14 (2 self)
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In this thesis, we develop constrained simulated annealing (CSA), a global optimization algorithm that asymptotically converges to constrained global minima (CGM dn ) with probability one, for solving discrete constrained nonlinear programming problems (NLPs). The algorithm is based on the necessary and sufficient condition for constrained local minima (CLM dn ) in the theory of discrete constrained optimization using Lagrange multipliers developed in our group. The theory proves the equivalence between the set of discrete saddle points and the set of CLM dn, leading to the firstorder necessary and sufficient condition for CLM dn. To find
Rigorous error bounds for the optimal value in semidefinite programming
 SIAM J. Numer. Anal
"... Abstract. A wide variety of problems in global optimization, combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with illconditioning and degeneracy, errone ..."
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Cited by 13 (4 self)
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Abstract. A wide variety of problems in global optimization, combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with illconditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library; these libraries contain many illconditioned and real life problems.
Globsol: History, composition, and advice on use
 In Global Optimization and Constraint Satisfaction, Lecture Notes in Computer Science
, 2003
"... Abstract. The GlobSol software package combines various ideas from interval analysis, automatic differentiation, and constraint propagation to provide verified solutions to unconstrained and constrained global optimization problems. After briefly reviewing some of these techniques and GlobSol’s deve ..."
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Cited by 10 (7 self)
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Abstract. The GlobSol software package combines various ideas from interval analysis, automatic differentiation, and constraint propagation to provide verified solutions to unconstrained and constrained global optimization problems. After briefly reviewing some of these techniques and GlobSol’s development history, we provide the first overall description of GlobSol’s algorithm. Giving advice on use, we point out strengths and weaknesses in GlobSol’s approaches. Through examples, we show how to configure and use GlobSol.
Discussion and Empirical Comparisons of Linear Relaxations and Alternate Techniques in Validated Deterministic Global Optimization
, 2004
"... VALIDATED GLOBAL OPTIMIZATION COMPARISONS 2 1 Introduction 1.1 The General Global Optimization Problem Our general global optimization problem can be stated as ..."
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Cited by 8 (0 self)
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VALIDATED GLOBAL OPTIMIZATION COMPARISONS 2 1 Introduction 1.1 The General Global Optimization Problem Our general global optimization problem can be stated as
Optimal Anytime Search For Constrained Nonlinear Programming
, 2001
"... In this thesis, we study optimal anytime stochastic search algorithms (SSAs) for solving general constrained nonlinear programming problems (NLPs) in discrete, continuous and mixedinteger space. The algorithms are general in the sense that they do not assume di#erentiability or convexity of functio ..."
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Cited by 6 (2 self)
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In this thesis, we study optimal anytime stochastic search algorithms (SSAs) for solving general constrained nonlinear programming problems (NLPs) in discrete, continuous and mixedinteger space. The algorithms are general in the sense that they do not assume di#erentiability or convexity of functions. Based on the search algorithms, we develop the theory of SSAs and propose optimal SSAs with iterative deepening in order to minimize their expected search time. Based on the optimal SSAs, we then develop optimal anytime SSAs that generate improved solutions as more search time is allowed. Our SSAs
Improved and simplified validation of feasible points: Inequality and equality constrained problems
 Mathematical Programming, submitted
, 2005
"... Abstract. In validated branch and bound algorithms for global optimization, upper bounds on the global optimum are obtained by evaluating the objective at an approximate optimizer; the upper bounds are then used to eliminate subregions of the search space. For constrained optimization, in general, a ..."
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Cited by 3 (1 self)
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Abstract. In validated branch and bound algorithms for global optimization, upper bounds on the global optimum are obtained by evaluating the objective at an approximate optimizer; the upper bounds are then used to eliminate subregions of the search space. For constrained optimization, in general, a small region must be constructed within which existence of a feasible point can be proven, and an upper bound on the objective over that region is obtained. We had previously proposed a perturbation technique for constructing such a region. In this work, we propose a much simplified and improved technique, based on an orthogonal decomposition of the normal space to the constraints. In purely inequality constrained problems, a point, rather than a region, can be used, and, for equality and inequality constrained problems, the region lies in a smallerdimensional subspace, giving rise to sharper upper bounds. Numerical experiments on published test sets for global optimization provide evidence of the superiority of the new approach within our GlobSol environment. 1.
Constraint aggregation for rigorous global optimization
"... Abstract In rigorous constrained global optimization, upper bounds on the objective function help to reduce the search space. Their construction requires finding a narrow box around an approximately feasible solution, verified to contain a feasible point. Approximations are easily found by local op ..."
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Abstract In rigorous constrained global optimization, upper bounds on the objective function help to reduce the search space. Their construction requires finding a narrow box around an approximately feasible solution, verified to contain a feasible point. Approximations are easily found by local optimization, but the verification often fails. In this paper we show that even if the verification of an approximate feasible point fails, the information extracted from the local optimization can still be used in many cases to reduce the search space. This is done by a rigorous filtering technique called constraint aggregation. It forms an aggregated redundant constraint, based on approximate Lagrange multipliers or on a vector valued measure of constraint violation. Using the optimality conditions, two sided linear relaxations, the GaussJordan algorithm and a directed modified Cholesky factorization, the information in the redundant constraint is turned into powerful bounds on the feasible set. Constraint aggregation is especially useful since it also works in a tiny neighborhood of the global optimizer, thereby reducing the cluster effect. A simple introductory example demonstrates how our new method works. Extensive tests show the performance on a large benchmark.
An Example of Singularity in Global Optimization
"... Abstract. Certain practical constrained global optimization problems have to date defied practical solution with interval branch and bound methods. The exact mechanism causing the difficulty has been difficult to pinpoint. Here, an example is given where the equality constraint set has higherorder ..."
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Abstract. Certain practical constrained global optimization problems have to date defied practical solution with interval branch and bound methods. The exact mechanism causing the difficulty has been difficult to pinpoint. Here, an example is given where the equality constraint set has higherorder singularities and degenerate manifolds of singularities on the feasible set. The reason that this causes problems is discussed, and ways of fixing it are suggested.