Month: July 2009

Band Gap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods

  • Robert M. Freund
  • Han Men
  • Ngoc-Cuong Nguyen
  • Pablo A. Parrilo
  • Jaime Peraire
    • Categories Optimization of Systems modeled by PDEs, Semi-definite Programming Tags , ,

    In this paper, we consider the optimal design of photonic crystal band structures for two-dimensional square lattices. The mathematical formulation of the band gap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector. To make the problem tractable, the original eigenvalue problem is discretized using … Read more

    An Augmented Lagrangian Approach for Sparse Principal Component Analysis

  • Zhaosong Lu
  • Yong Zhang
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization, Statistics Tags , , ,

    Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To … Read more

    Old Wine in a New Bottle: The MILP Road to MIQCP

  • Samuel Burer
  • Anureet Saxena
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Global Optimization Theory Tags , , ,

    This paper surveys results on the NP-hard mixed-integer quadratically constrained programming problem. The focus is strong convex relaxations and valid inequalities, which can become the basis of efficient global techniques. In particular, we discuss relaxations and inequalities arising from the algebraic description of the problem as well as from dynamic procedures based on disjunctive programming. … Read more

    Sample Average Approximation for Stochastic Dominance Constrained Programs

  • Tito Homem-de-Mello
  • Jian Hu
  • Sanjay Mehrotra
    • Categories Nonlinear Optimization, Stochastic Programming Tags , , , , ,

    In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems has been receiving increasing attention in the literature as it allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with … Read more

    Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema

  • Vladislav B. Tadic
    • Categories Control Applications, Stochastic Programming Tags , , , , ,

    The asymptotic behavior of stochastic gradient algorithms is studied. Relying on some results of differential geometry (Lojasiewicz gradient inequality), the almost sure point-convergence is demonstrated and relatively tight almost sure bounds on the convergence rate are derived. In sharp contrast to all existing result of this kind, the asymptotic results obtained here do not require … Read more

    The Farkas Lemma Revisited

  • S. S. Kutateladze
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    The Farkas Lemma is extended to simultaneous linear operator and polyhedral sublinear operator inequalities by Boolean valued analysis. CitationSobolev Institute of Mathematics, Novosibirsk, 630090 RussiaArticleDownload View PDF

    A Combined Class of Self-Scaling and Modified Quasi-Newton Methods

  • Mehiddin Al-Baali
  • Humaid Khalfan
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , ,

    Techniques for obtaining safely positive definite Hessian approximations with self-scaling and modified quasi-Newton updates are combined to obtain `better’ curvature approximations in line search methods for unconstrained optimization. It is shown that this class of methods, like the BFGS method has global and superlinear convergence for convex functions. Numerical experiments with this class, using the … Read more