Nonlinear Optimization

Newton–Picard-Based Preconditioning for Linear-Quadratic Optimization Problems with Time-Periodic Parabolic PDE Constraints

  • Hans Georg Bock
  • Andreas Potschka
  • M.S. Mommer
  • Johannes P. Schlöder
    • Categories Optimization of Systems modeled by PDEs, Quadratic Programming Tags , , , ,

    We develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear-quadratic optimization problems with time-periodic parabolic PDE constraints. The resulting real-valued linear system to be solved is symmetric indefinite. We propose all-at-once symmetric indefinite preconditioners based on a Newton–Picard approach which divides the variable space into slow … Read more

    Estimating Computational Noise

  • Jorge Moré
  • Stefan M. Wild
    • Categories Nonlinear Optimization, Optimization Software and Modeling Systems, Other Topics Tags , ,

    Computational noise in deterministic simulations is as ill-defined a concept as can be found in scientific computing. When coupled with adaptive strategies, the effects of finite precision destroy smoothness of the simulation output and complicate subsequent analysis. Following the work of Hamming on roundoff errors, we present a new algorithm, ECnoise, for quantifying the noise … Read more

    A Gauss-Newton approach for solving constrained optimization problems using differentiable exact penalties

  • Roberto Andreani
  • Ellen H. Fukuda
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We propose a Gauss-Newton-type method for nonlinear constrained optimization using the exact penalty introduced recently by Andre and Silva for variational inequalities. We extend their penalty function to both equality and inequality constraints using a weak regularity assumption, and as a result, we obtain a continuously differentiable exact penalty function and a new reformulation of … Read more

    An interior-point method for minimizing the sum of piecewise-linear convex functions

  • Toshihiro Kosaki
    • Categories Nonlinear Optimization Tags , ,

    We consider the problem to minimize the sum of piecewise-linear convex functions under both linear and nonnegative constraints. We convert the piecewise-linear convex problem into a standard form linear programming problem (LP) and apply a primal-dual interior-point method for the LP. From the solution of the converted problem, we can obtain the solution of the … Read more

    A concave optimization-based approach for sparse portfolio selection

  • David Di Lorenzo
  • Giampaolo Liuzzi
  • Francesco Rinaldi
  • Fabio Schoen
  • Marco Sciandrone
    • Categories Constrained Nonlinear Optimization, Finance and Economics, Stochastic Approaches Tags , , ,

    This paper considers a portfolio selection problem in which portfolios with minimum number of active assets are sought. This problem is motivated by the need of inducing sparsity on the selected portfolio to reduce transaction costs, complexity of portfolio management, and instability of the solution. The resulting problem is a difficult combinatorial problem. We propose … Read more

    ^phBcnorms, log-barriers and Cramer transform in optimization

  • Jean-Bernard Lasserre
  • Eduardo Santillan Zeron
    • Categories Constrained Nonlinear Optimization, Convex Optimization

    We show that the Laplace approximation of a supremum by $L^p$-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem $P$ and its dual $P^*$ appear naturally when using this simple approximation technique for the value function $g$ of $P$ or its Legendre-Fenchel conjugate $g^*$. In addition, … Read more

    A Robust Implementation of a Sequential Quadratic Programming Algorithm with Successive Error Restoration

  • Klaus Schittkowski
    • Categories Constrained Nonlinear Optimization

    We consider sequential quadratic programming (SQP) methods for solving constrained nonlinear programming problems. It is generally believed that SQP methods are sensitive to the accuracy by which partial derivatives are provided. One reason is that differences of gradients of the Lagrangian function are used for updating a quasi-Newton matrix, e.g., by the BFGS formula. The … Read more

    A Feasible Directions Method for Nonsmooth Convex Optimization

  • Wilhelm Freire
  • Jose Herskovits
  • Mario Tanaka Fo.
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization

    We propose a new technique for minimization of convex functions not necessarily smooth. Our approach employs an equivalent constrained optimization problem and approximated linear programs obtained with cutting planes. At each iteration a search direction and a step length are computed. If the step length is considered “non serious”, a cutting plane is added and … Read more

    On Duality Gap in Binary Quadratic Programming

  • Jianjun Gao
  • Duan Li
  • Chunli Liu
  • Xiaoling Sun
    • Categories 0-1 Programming, Quadratic Programming, Semi-definite Programming

    We present in this paper new results on the duality gap between the binary quadratic optimization problem and its Lagrangian dual or semidefinite programming relaxation. We first derive a necessary and sufficient condition for the zero duality gap and discuss its relationship with the polynomial solvability of the primal problem. We then characterize the zeroness … Read more

    On the convergence of a wide range of trust region methods for unconstrained optimization

  • M.J.D. Powell
    • Categories Unconstrained Optimization

    We consider trust region methods for seeking the unconstrained minimum of an objective function F(x), x being the vector of variables, when the gradient grad F is available. The methods are iterative with a starting point x_1 being given. The new vector of variables x_(k+1) is derived from a quadratic approximation to F that interpolates … Read more