In this paper we consider a practical variation of the Cheney-Goldstein-Kelley cutting plane algorithm which solves linear relaxations using the lexicographic dual simplex method and allows for the deletion of inactiv...
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In this paper we consider a practical variation of the Cheney-Goldstein-Kelley cutting plane algorithm which solves linear relaxations using the lexicographic dual simplex method and allows for the deletion of inactive cuts. The algorithm presented can be shown to exhibit a geometric global rate of convergence asymptotically under mild assumptions. Other proofs of the rate of convergence for this type of algorithm require restrictive assumptions such as strict convexity of the objective function or a Haar condition. The approach taken here requires only that at each iteration of the algorithm a cut is produced for each convex constraint.
The following problem is studied: Given a compact setS inR n and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R n satisfyingp(y?x) ≤ r is...
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The following problem is studied: Given a compact setS inR n and a Minkowski functionalp(x), find the largest positive numberr for which there existsx ∈ S such that the set of ally ∈ R n satisfyingp(y?x) ≤ r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this “design centering problem” can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR n . In the case where, moreover,p(x) = (x T Ax)1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.
We will present a new method for finding the global minimum of a Lipschitzian function under Lipschitzian constraints. The method consists in converting the given problem into one of globally minimizing a concave func...
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This paper presents a new class of outerapproximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. ...
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This paper presents a new class of outerapproximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. Moreover, the methods employ quadratic objective functions in the subproblems by adding a simple quadratic term to the objective function of the original problem, while other outerapproximation methods usually use the original objective function itself throughout the iterations. By this modification, convergence of the methods can be proved under mild conditions. Furthermore, it is shown that generalized versions of the cut construction schemes in Kelley-Cheney-Goldstein's cutting plane method and Veinott's supporting hyperplane method can be incorporated with the present methods and a cut generated at each iteration need not be retained in the succeeding iterations.
This paper presents an implementable algorithm of the outerapproximations type for solving nonlinear programming problems with functional inequality constraints. The algorithm was motivated by engineering design prob...
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This paper presents an implementable algorithm of the outerapproximations type for solving nonlinear programming problems with functional inequality constraints. The algorithm was motivated by engineering design problems in circuit tolerancing, multivariable control, and shock-resistant structures.
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