15,672 research outputs found

    An Algorithm for a Class of Nonconvex Programming Problems

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    Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

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    We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximization for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimization problems with the help of convex semidefinite programming (optimization over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach

    A deterministic-stochastic method for nonconvex MINLP problems

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    A mixed-integer programming problem is one where some of the variables must have only integer values. Although some real practical problems can be solved with mixed-integer linear methods, there are problems occurring in the engineering area that are modelled as mixed-integer nonlinear programming (MINLP) problems. When they contain nonconvex functions then they are the most difficult of all since they combine all the difficulties arising from the two sub-classes: mixed-integer linear programming and nonconvex nonlinear programming (NLP). Efficient deterministic methods for solving MINLP are clever combinations of Branch-and-Bound (B&B) and Outer-Approximations classes. When solving nonconvex NLP relaxation problems that arise in the nodes of a tree in a B&B algorithm, using local search methods, only convergence to local optimal solutions is guaranteed. Pruning criteria cannot be used to avoid an exhaustive search in the solution space. To address this issue, we propose the use of a simulated annealing algorithm to guarantee convergence, at least with probability one, to a global optimum of the nonconvex NLP relaxation problem. We present some preliminary tests with our algorithm.Fundação para a CiĂȘncia e a Tecnologi

    A unified branch-and-bound and cutting plane algorithm for a class of nonconvex optimization problems : application to bilinear programming

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    A unifled approach to branch-and-bound and cutting plane methods for solving a certain class of nonconvex optimization problems is proposed. Based on this approach an implementable algorithm is obtained for programming problems with a bilinear objective function and jointly convex constraints

    A second derivative SQP method: theoretical issues

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    Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be computationally nonviable. This paper presents a second-derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally. Additionally, an explicit descent-constraint is imposed on certain QP subproblems, which “guides” the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established

    Algorithm for solution of convex MINLP problems

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    The current work shows the fonnulation and implementation of an algorithm for the solution of convex rnixed-integer nonlinear programming (MINLP) problems. The proposed algorithm does not folJow the traditional sequence of solutions of nonlinear programming (NLP) subproblems and master mixed-integer linear programming (MILP) problems. lnstead, the mas ter problem is defined dynamically during the tree search to reduce the number of nodes that need to be enumerated. A branch and bound search is perfonned to predict lower bound by solving linear programrning (LP) subproblems until feasible integer solutions are found. For these nades, noolinear programming subproblems are olved, providing upper bounds and new linear approximations, which are used to tighten the linear representation of the open nodes in the search tree. Numerical results for convex and nonconvex test problems are analyzed, comparing the efficiency of the proposed algorithm and the general algebraic modeling system (GAMS)
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