Nonlinear Optimization

On the Convergence of Successive Linear-Quadratic Programming Algorithms

  • Richard H. Byrd
  • Nicholas I. M. Gould
  • Jorge Nocedal
  • Richard A. Waltz
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    The global convergence properties of a class of penalty methods for nonlinear programming are analyzed. These methods include successive linear programming approaches, and more specifically, the successive linear-quadratic programming approach presented by Byrd, Gould, Nocedal and Waltz (Math. Programming 100(1):27–48, 2004). Every iteration requires the solution of two trust-region subproblems involving piecewise linear and quadratic … Read more

    The Q Method for Second-order Cone Programming

  • Farid Alizadeh
  • Yu Xia
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming, Semi-definite Programming Tags , ,

    Based on the Q method for SDP, we develop the Q method for SOCP. A modified Q method is also introduced. Properties of the algorithms are discussed. Convergence proofs are given. Finally, we present numerical results. CitationAdvOl-Report#2004/15 McMaster University, Advanced Optimization LaboratoryArticleDownload View PDF

    A New Conjugate Gradient Algorithm Incorporating Adaptive Ellipsoid Preconditioning

  • Renato D.C. Monteiro
  • Arkadi Nemirovski
  • Jerome W. O'Neal
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , , , , ,

    The conjugate gradient (CG) algorithm is well-known to have excellent theoretical properties for solving linear systems of equations $Ax = b$ where the $n\times n$ matrix $A$ is symmetric positive definite. However, for extremely ill-conditioned matrices the CG algorithm performs poorly in practice. In this paper, we discuss an adaptive preconditioning procedure which improves the … Read more

    A survey of the S-lemma

  • Imre Pólik
  • Tamás Terlaky
    • Categories Control Applications, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , , , , ,

    In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry and linear algebra as well. These were active research areas, but as … Read more

    Newton-KKT Interior-Point Methods for Indefinite Quadratic Programming

  • Pierre-Antoine Absil
  • André L. Tits
    • Categories Constrained Nonlinear Optimization, Quadratic Programming Tags , , ,

    Two interior-point algorithms are proposed and analyzed, for the (local) solution of (possibly) indefinite quadratic programming problems. They are of the Newton-KKT variety in that (much like in the case of primal-dual algorithms for linear programming) search directions for the `primal´ variables and the Karush-Kuhn-Tucker (KKT) multiplier estimates are components of the Newton (or quasi-Newton) … Read more

    Nonlinear-Programming Reformulation of the Order-Value Optimization problem

  • Roberto Andreani
  • José Mario Martínez
  • Cibele Dunder
    • Categories Nonlinear Optimization Tags , , , ,

    Order-value optimization (OVO) is a generalization of the minimax problem motivated by decision-making problems under uncertainty and by robust estimation. New optimality conditions for this nonsmooth optimization problem are derived. An equivalent mathematical programming problem with equilibrium constraints is deduced. The relation between OVO and this nonlinear-programming reformulation is studied. Particular attention is given to … Read more

    A shifted Steihaug-Toint method for computing a trust-region step.

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Unconstrained Optimization Tags , , , , ,

    Trust-region methods are very convenient in connection with the Newton method for unconstrained optimization. The More-Sorensen direct method and the Steihaug-Toint iterative method are most commonly used for solving trust-region subproblems. We propose a method which combines both of these approaches. Using the small-size Lanczos matrix, we apply the More-Sorensen method to a small-size trust-region … Read more

    Augmented Lagrangian methods under the Constant Positive Linear Dependence constraint qualification

  • Roberto Andreani
  • Ernesto G. Birgin
  • José Mario Martínez
  • María Laura Schuverdt
    • Categories Nonlinear Optimization Tags , , ,

    Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under … Read more

    SENSITIVITY ANALYSIS IN CONVEX QUADRATIC OPTIMIZATION: INVARIANT SUPPORT SET INTERVAL

  • Alireza Ghaffari Hadigheh
  • Tamás Terlaky
    • Categories Quadratic Programming

    In sensitivity analysis one wants to know how the problem and the optimal solutions change under the variation of the input data. We consider the case when variation happens in the right hand side of the constraints and/or in the linear term of the objective function. We are interested to find the range of the … Read more

    Performance of CONDOR, a Parallel, Constrained extension of Powell’s UOBYQA algorithm. Experimental results and comparison with the DFO algorithm.

  • Hugues Bersini
  • Frank Vanden Berghen
    • Categories Constrained Nonlinear Optimization, Optimization of Systems modeled by PDEs, Parallel Algorithms Tags , , ,

    This paper presents an algorithmic extension of Powell’s UOBYQA algorithm (”Unconstrained Optimization BY Quadratical Approximation”). We start by summarizing the original algorithm of Powell and by presenting it in a more comprehensible form. Thereafter, we report comparative numerical results between UOBYQA, DFO and a parallel, constrained extension of UOBYQA that will be called in the … Read more