2,235 research outputs found
Hybrid Newton-type method for a class of semismooth equations
In this paper, we present a hybrid method for the solution of a class of composite semismooth equations encountered frequently in applications. The method is obtained by combining a generalized finite-difference Newton method to an inexpensive direct search method. We prove that, under standard assumptions, the method is globally convergent with a local rate of convergence which is superlinear or quadratic. We report also several numerical results obtained applying the method to suitable reformulations of well-known nonlinear complementarity problem
Linear complementarity problems on extended second order cones
In this paper, we study the linear complementarity problems on extended
second order cones. We convert a linear complementarity problem on an extended
second order cone into a mixed complementarity problem on the non-negative
orthant. We state necessary and sufficient conditions for a point to be a
solution of the converted problem. We also present solution strategies for this
problem, such as the Newton method and Levenberg-Marquardt algorithm. Finally,
we present some numerical examples
A second derivative SQP method: local convergence
In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact ā1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud
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Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bkāa simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ā1-penalty function over a sequence of increasing values of the penalty parameter.\ud
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Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set
An interior point algorithm for computing equilibria in economies with incomplete asset markets
Computing equilibria in general equilibria models with incomplete asset (GEI) markets is technically difficult. The standard numerical methods for computing these equilibria are based on homotopy methods. Despite recent advances in computational economics, much more can be done to enlarge the catalogue of techniques for computing GEI equilibria. This paper presents an interior-point algorithm that exploits the special structure of GEI markets. We prove that the algorithm converges globally at a quadratic rate, rendering it particularly effective in solving large-scale GEI economies. To illustrate its performance, we solve relevant examples of GEI market
AN INTERIOR POINT ALGORITHM FOR COMPUTING EQUILIBRIA IN ECONOMIES WITH INCOMPLETE ASSET MARKETS
Computing equilibria in general equilibria models with incomplete asset (GEI) markets is technically difficult. The standard numerical methods for computing these equilibria are based on homotopy methods. Despite recent advances in computational economics, much more can be done to enlarge the catalogue of techniques for computing GEI equilibria. This paper presents an interior-point algorithm that exploits the special structure of GEI markets. We prove that the algorithm converges globally at a quadratic rate, rendering it particularly effective in solving large-scale GEI economies. To illustrate its performance, we solve relevant examples of GEI markets.
A sequential semidefinite programming method and an application in passive reduced-order modeling
We consider the solution of nonlinear programs with nonlinear
semidefiniteness constraints. The need for an efficient exploitation of the
cone of positive semidefinite matrices makes the solution of such nonlinear
semidefinite programs more complicated than the solution of standard nonlinear
programs. In particular, a suitable symmetrization procedure needs to be chosen
for the linearization of the complementarity condition. The choice of the
symmetrization procedure can be shifted in a very natural way to certain linear
semidefinite subproblems, and can thus be reduced to a well-studied problem.
The resulting sequential semidefinite programming (SSP) method is a
generalization of the well-known SQP method for standard nonlinear programs. We
present a sensitivity result for nonlinear semidefinite programs, and then
based on this result, we give a self-contained proof of local quadratic
convergence of the SSP method. We also describe a class of nonlinear
semidefinite programs that arise in passive reduced-order modeling, and we
report results of some numerical experiments with the SSP method applied to
problems in that class
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