Constrained Nonlinear Optimization

Nonlinear Model Predictive Control via Feasibility-Perturbed Sequential Quadratic Programming

  • J B Rawlings
  • M. J. Tenny
  • Stephen Wright
    • Categories Chemical Engineering, Constrained Nonlinear Optimization, Control Applications Tags , ,

    Model predictive control requires the solution of a sequence of continuous optimization problems that are nonlinear if a nonlinear model is used for the plant. We describe briefly a trust-region feasibility-perturbed sequential quadratic programming algorithm (developed in a companion report), then discuss its adaptation to the problems arising in nonlinear model predictive control. Computational experience … Read more

    A Simplified/Improved HKM Direction for Certain Classes of Semidefinite Programming

  • Franz Rendl
  • Renata Sotirov
  • Henry Wolkowicz
    • Categories Constrained Nonlinear Optimization, Semi-definite Programming Tags , , ,

    Semidefinite Programming (SDP) provides strong bounds for many NP-hard combinatorial problems. Arguably the most popular/efficient search direction for solving SDPs using a primal-dual interior point (p-d i-p) framework is the {\em HKM direction}. This direction is a Newton direction found from the linearization of a symmetrized version of the optimality conditions. For many of the … Read more

    Interior-Point Methods for Nonconvex Nonlinear Programming: Complementarity Constraints

  • Hande Benson
  • David F. Shanno
  • Robert J. Vanderbei
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    In this paper, we present the formulation and solution of optimization problems with complementarity constraints using an interior-point method for nonconvex nonlinear programming. We identify possible difficulties that could arise, such as unbounded faces of dual variables, linear dependence of constraint gradients and initialization issues. We suggest remedies. We include encouraging numerical results on the … Read more

    A Primal-Dual Interior-Point Method for Nonlinear Programming with Strong Global and Local Convergence Properties.

  • Sasan Bakhtiari
  • Craig T. Lawrence
  • André L. Tits
  • Thomas J. Urban
  • Andreas Wächter
    • Categories Constrained Nonlinear Optimization Tags , ,

    An exact-penalty-function-based scheme—inspired from an old idea due to Mayne and Polak (Math. Prog., vol.~11, 1976, pp.~67–80)—is proposed for extending to general smooth constrained optimization problems any given feasible interior-point method for inequality constrained problems. It is shown that the primal-dual interior-point framework allows for a simpler penalty parameter update rule than that discussed and … Read more

    A new exact penalty function

  • Waltraud Huyer
  • Arnold Neumaier
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    For constrained smooth or nonsmooth optimization problems, new continuously differentiable penalty functions are derived. They are proved exact in the sense that under some nondegeneracy assumption, local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function. This is achieved by augmenting the dimension of the program by a variable that … Read more

    Combinatorial Structures in Nonlinear Programming

  • Stefan Scholtes
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization, Robust Optimization Tags , , ,

    Non-smoothness and non-convexity in optimization problems often arise because a combinatorial structure is imposed on smooth or convex data. The combinatorial aspect can be explicit, e.g. through the use of ”max”, ”min”, or ”if” statements in a model, or implicit as in the case of bilevel optimization where the combinatorial structure arises from the possible … Read more

    The Penalty Interior Point Method fails to converge for Mathematical Programs with Equilibrium Constraints

  • Sven Leyffer
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , ,

    This paper presents a small example for which the Penalty Interior Point Method converges to a non-stationary point. The reasons for this adverse behaviour are discussed. Citation Numerical Analysis Report NA/208, Department of Mathematics, University of Dundee, February 2002. Article Download View The Penalty Interior Point Method fails to converge for Mathematical Programs with Equilibrium … Read more

    Local convergence of SQP methods for Mathematical Programs with Equilibrium Constraints

  • Roger Fletcher
  • Sven Leyffer
  • Daniel Ralph
  • Stefan Scholtes
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , ,

    Recently, it has been shown that Nonlinear Programming solvers can successfully solve a range of Mathematical Programs with Equilibrium Constraints (MPECs). In particular, Sequential Quadratic Programming (SQP) methods have been very successful. This paper examines the local convergence properties of SQP methods applied to MPECs. It is shown that SQP converges superlinearly under reasonable assumptions … Read more

    Relations between divergence of multipliers and convergence to infeasible points in primal-dual interior methods for nonconvex nonlinear programming

  • Anders Forsgren
  • Göran Sporre
    • Categories Constrained Nonlinear Optimization Tags , , ,

    Recently, infeasibility issues in interior methods for nonconvex nonlinear programming have been studied. In particular, it has been shown how many line-search interior methods may converge to an infeasible point which is on the boundary of the feasible region with respect to the inequality constraints. The convergence is such that the search direction does not … Read more

    Reduntant axioms in the definitionof Bregman functions

  • Dan Butnariu
  • Charles Byrne
  • Yair Censor
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    The definition of a Bregman function, given by Censor and Lent in 1981 on the basis of Bregman’s seminal 1967 paper, was subsequently used in a plethora of research works as a tool for building sequential and inherently parallel feasibility and optimization algorithms. Solodov and Svaiter have recently shown that it is not Citation Journal … Read more