Browsing by Author "Dennis, J.E."

Now showing 1 - 8 of 8
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    A Curvilinear Search Using Tridiagonal Secant Updates for Unconstrained Optimization
    (1990-03) Dennis, J.E.; Echebest, N.; Guardarucci, M.T.; Martinez, J.M.; Scolnik, H.D.; Vaccino, C.
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    Direct Search Methods on Parallel Machines
    (1990-09) Dennis, J.E.; Torczon, Virginia
    This paper describes an approach to constructing derivative-free parallel algorithms for unconstrained optimization which are easy to implement on parallel machines. A special feature of this approach is the ease with which algorithms can be generated to take advantage of any number of processors and to adapt to any cost ratio of communication to function evaluation. The algorithms given here are supported by a strong convergence theorem, promising computational results, and an intuitively appealing interpretation as multi-directional line search methods.
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    On Alternative Problem Formulations for Multidisciplinary Design Optimization
    (1992-12) Cramer, E.J.; Frank, P.D.; Shubin, G.R.; Dennis, J.E.; Lewis, R.M.
    In this paper we introduce a perspective on multidisciplinary design optimization (MDO) problem formulation that provides a basis for choosing among existing formulations and suggests provocative, new ones. MDO problems offer a richer spectrum of possibilities for problem formulation than do single discipline design optimization problems, or multidisciplinary analysis problems. This is because the variables and the equations that characterize the MDO problem can be "partitioned" in some interesting ways between what we traditionally think of as the "analysis code(s)" and the "optimization code." An MDO approach can be characterized by what part of the overall computation is done in each code, how that computation is done, and what information is communicated between the codes.
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    On the Superlinear and Quadratic Convergence of Primal-Dual Interior Point Linear Programming Algorithms
    (1990-01) Zhang, Yin; Tapia, R.A.; Dennis, J.E.
    This paper presents a convergence rate analysis for interior point primal-dual linear programming algorithms. Conditions that guarantee Q-superlinear convergence are identified in two distinct theories. Both state that, under appropriate assumptions, Q-superlinear convergence is achieved by asymptotically taking the step to the boundary of the positive orthant and letting the barrier parameter approach zero at a rate that is superlinearly faster than the convergence of the duality gap to zero. The first theory makes no nondegeneracy assumption and explains why in recent numerical experimentation Q-superlinear convergence was always observed. The second theory requires the restrictive assumption of primal nondegeneracy. However, it gives the surprising result that Q-superlinear convergence can still be attained even if centering is not phased out, provided the iterates asymptotically approach the central path. The latter theory is extended to produce a satisfactory Q-quadratic convergence theory. It requires that the step approach the boundary as fast as the duality gap approaches zero and the barrier parameter approach zero as fast as the square of the duality gap approaches zero.
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    On the Superlinear Convergence of Interior Point Algorithms for a General Class of Problems
    (1990-03) Zhang, Yin; Tapia, R.A.; Dennis, J.E.
    In this paper, we extend the Q-superlinear convergence theory recently developed by Zhang, Tapia and Dennis for a class of interior point linear programming algorithms to similar interior point algorithms for quadratic programming and for linear complementarity problems. Our unified approach consists of viewing all these algorithms as the damped Newton method applied to perturbations of a general problem. We show that under appropriate assumptions, Q-superlinear convergence can be achieved by asymptotically taking the step to the boundary of the positive orthant and letting the barrier (or path-following) parameter approach zero at a specific rate.
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    Sizing and Least Change Secant Methods
    (1990-03) Dennis, J.E.; Wolkowicz, H.
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    Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems
    (1994-12) Dennis, J.E.; Heinkenschloss, Matthias; Vicente, Luís N.
    In this paper a family of trust-region interior-point SQP algorithms for the solution of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using a primal-dual affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trust-region techniques for equality-constrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques. Global convergence of these algorithms to a first-order KKT limit point is proved under very mild conditions on the trial steps. Under reasonable, but more stringent conditions on the quadratic model and on the trial steps, the sequence of iterates generated by the algorithms is shown to have a limit point satisfying the second-order necessary KKT conditions. The local rate of convergence to a nondegenerate strict local minimizer is q-quadratic. The results given here include as special cases current results for only equality constraints and for only simple bounds. Numerical results for the solution of an optimal control problem governed by a nonlinear heat equation are reported.