Constrained Nonlinear Optimization

Extension of Quasi-Newton Methods to Mathematical Programs with Complementarity Constraints

  • Houyuan Jiang
  • Daniel Ralph
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , , ,

    Quasi-Newton methods in conjunction with the piecewise sequential quadratic programming are investigated for solving mathematical programming with equilibrium constraints, in particular for problems with complementarity constraints. Local convergence as well as superlinear convergence of these quasi-Newton methods can be established under suitable assumptions. In particular, several well-known quasi-Newton methods such as BFGS and DFP are … Read more

    Convergence of a Penalty Method for Mathematical Programmingwith ComplementarityConstraints

  • Xinmin Hu
  • Daniel Ralph
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , ,

    We adapt the convergence analysis of smoothing (Fukushima and Pang) and regularization (Scholtes) methods to a penalty framework for mathematical programs with complementarity constraints (MPCCs), and show that the penalty framework shares similar convergence properties to these methods. Moreover, we give sufficient conditions for a sequence generated by the penalty framework to be attracted to … Read more

    Numerical experience with solving MPECs as NLPs

  • Roger Fletcher
  • Sven Leyffer
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Optimization Software Benchmark Tags , , , ,

    This paper describes numerical experience with solving MPECs as NLPs on a large collection of test problems. The key idea is to use off-the-shelf NLP solvers to tackle large instances of MPECs. It is shown that SQP methods are very well suited to solving MPECs and at present outperform Interior Point solvers both in terms … Read more

    A Feasible Trust-Region Sequential Quadratic Programming Algorithm

  • M. J. Tenny
  • Stephen Wright
    • Categories Constrained Nonlinear Optimization Tags , ,

    An algorithm for smooth nonlinear constrained optimization problems is described, in which a sequence of feasible iterates is generated by solving a trust-region sequential quadratic programming (SQP) subproblem at each iteration, and perturbing the resulting step to retain feasibility of each iterate. By retaining feasibility, the algorithm avoids several complications of other trust-region SQP approaches: … Read more

    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