Nonlinear Optimization

Sensitivity analysis in convex quadratic optimization: simultaneous perturbation of the objective and right-hand-side vectors

  • Alireza Ghaffari Hadigheh
  • Oleksandr Romanko
  • Tamás Terlaky
    • Categories Finance and Economics, Linear Programming, Quadratic Programming Tags , , , , , ,

    In this paper we study the behavior of Convex Quadratic Optimization problems when variation occurs simultaneously in the right-hand side vector of the constraints and in the coefficient vector of the linear term in the objective function. It is proven that the optimal value function is piecewise-quadratic. The concepts of transition point and invariancy interval … Read more

    On the control of an evolutionary equilibrium in micromagnetics

  • Michal Kočvara
  • Martin Kružík
  • Jiri V. Outrata
    • Categories Basic Sciences Applications, Complementarity and Variational Inequalities, Systems governed by Differential Equations Optimization Tags , , , ,

    We formulate an optimal control problem of magnetization in a ferromagnet as a mathematical program with evolutionary equilibrium constraints. The evolutionary nature of the equilibrium is due to the hysteresis behavior of the respective magnetization process. To solve the problem numerically, we adapted the implicit programming technique. The adjoint equations, needed to compute the subgradients … Read more

    An Optimization Approach to Computing the Implied Volatility of American Options

  • Arun Sen
    • Categories Complementarity and Variational Inequalities, Finance and Economics, Nonlinear Optimization Tags , ,

    We present a method to compute the implied volatility of American options as a mathematical program with equilibrium constraints. The formulation we present is new, as are the convergence results we prove. The algorithm holds the promise of being practical to implement, and we demonstrate some preliminary numerical results to this end. CitationPrinceton University working … Read more

    Active Set Identification in Nonlinear Programming

  • Christina Oberlin
  • Stephen Wright
    • Categories Constrained Nonlinear Optimization Tags , ,

    Techniques that identify the active constraints at a solution of a nonlinear programming problem from a point near the solution can be a useful adjunct to nonlinear programming algorithms. They have the potential to improve the local convergence behavior of these algorithms, and in the best case can reduce an inequality constrained problem to an … Read more

    A generating set search method exploiting curvature and sparsity

  • Lennart Frimannslund
  • Trond Steihaug
    • Categories Unconstrained Optimization Tags , , , ,

    Generating Set Search method are one of the few alternatives for optimising high fidelity functions with numerical noise. These methods are usually only efficient when the number of variables is relatively small. This paper presents a modification to an existing Generating Set Search method, which makes it aware of the sparsity structure of the Hessian. … Read more

    On the Global Convergence of a Trust Region Method for Solving Nonlinear Constraints Infeasibility Problem

  • Choong Ming Chin
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonlinear Systems and Least-Squares Tags , , , ,

    A framework for proving global convergence for a class of nonlinear constraints infeasibility problem is presented without assuming that the Jacobian has full rank everywhere. The underlying method is based on the simple sufficient reduction criteria where trial points are accepted provided there is a sufficient decrease in the constraints violation function. The proposed methods … Read more

    Set Intersection Theorems and Existence of Optimal Solutions

  • Dimitri Bertsekas
  • Paul Tseng
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , ,

    The question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics, including the existence of optimal solutions, the validity of the minimax inequality in zero sum games, and the absence of a duality gap in constrained optimization. We introduce the new notion of … Read more

    Interior Methods for Mathematical Programs with Complementarity Constraints

  • Sven Leyffer
  • Gabriel Lopez-Calva
  • Jorge Nocedal
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    This paper studies theoretical and practical properties of interior-penalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then … Read more

    Steering Exact Penalty Methods for Optimization

  • Richard H. Byrd
  • Jorge Nocedal
  • Richard A. Waltz
    • Categories Nonlinear Optimization Tags ,

    This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. In contrast with classical approaches, the choice of the penalty parameter ceases to be … Read more