New Fractional Error Bounds for Nonconvex Polynomial Systems with Applications to Holderian Stability in Optimization and Spectral Theory of Tensors

  • Guoyin Li
  • Boris S. Mordukhovich
  • Tiên-So'n Pham
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    In this paper we derive new fractional error bounds for nonconvex polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. The results obtained do not require any regularity assumptions and resolve, in particular, some open questions posed in the literature. The developed techniques are … Read more

    Spectral bounds for the independence ratio and the chromatic number of an operator

  • Christine Bachoc
  • Fernando Mário de Oliveira Filho
  • Frank Vallentin
  • Evan DeCorte
    • Categories Linear, Cone and Semidefinite Programming Tags , , , , ,

    We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical … Read more

    VSDP: A Matlab toolbox for verified semidefinite-quadratic-linear programming

  • Viktor Härter
  • Christian Jansson
  • Marko Lange
    • Categories Linear, Cone and Semidefinite Programming, Optimization Software and Modeling Systems Tags , , , ,

    VSDP is a software package that is designed for the computation of verified results in conic programming. The current version of VSDP supports the constraint cone consisting of the product of semidefinite cones, second-order cones and the nonnegative orthant. It provides functions for computing rigorous error bounds of the true optimal value, verified enclosures of … Read more

    Transmission Expansion Planning Using an AC Model: Formulations and Possible Relaxations

  • G. Th. Heydt
  • Hans D Mittelmann
  • Vijay Vittal
  • Hui Zhang
    • Categories Facility Planning and Design Tags , , , , ,

    Transmission expansion planning (TEP) is a rather complicated process which requires extensive studies to determine when, where and how many transmission facilities are needed. A well planned power system will not only enhance the system reliability, but also tend to contribute positively to the overall system operating efficiency. Starting with two mixed-integer nonlinear programming (MINLP) … Read more

    Second-Order Variational Analysis in Conic Programming with Applications to Optimality and Stability

  • Boris S. Mordukhovich
  • Jiri V. Outrata
  • Héctor Ramírez
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized di erential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related … Read more

    A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold

  • Yu-Hong Dai
  • Bo Jiang
    • Categories Applications - Science and Engineering, Nonlinear Optimization Tags , , , , , , , , , , ,

    This paper considers optimization problems on the Stiefel manifold $X^TX=I_p$, where $X\in \mathbb{R}^{n \times p}$ is the variable and $I_p$ is the $p$-by-$p$ identity matrix. A framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of $X$ and the null space of $X^T$. While this general framework … Read more

    Risk-Averse Control of Undiscounted Transient Markov Models

  • Özlem Cavus
  • Andrzej Ruszczyński
    • Categories Dynamic Programming, Stochastic Programming

    We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost problem for a transient controlled Markov process. We derive risk-averse dynamic programming equations and we show that a randomized policy may be strictly better than deterministic policies, when risk measures are employed. We illustrate the results on an optimal stopping … Read more

    A branch-and-bound algorithm for biobjective mixed-integer programs

  • Pietro Belotti
  • Banu Soylu
  • Margaret Wiecek
    • Categories (Mixed) Integer Linear Programming, Multi-Criteria Optimization Tags , , ,

    We propose a branch-and-bound (BB) algorithm for biobjective mixed-integer linear programs (BOMILPs). Our approach makes no assumption on the type of problem and we prove that it returns all Pareto points of a BOMILP. We discuss two techniques upon which the BB is based: fathoming rules to eliminate those subproblems that are guaranteed not to … Read more

    Partial Second-Order Subdifferentials in Variational Analysis and Optimization

  • Boris S. Mordukhovich
  • Nguyen Mau Nam
  • Nguyen Thi Yen Nhi
    • Categories Nonsmooth Optimization Tags , , , , , ,

    This paper presents a systematic study of partial second-order subdifferentials for extended-real-valued functions, which have already been applied to important issues of variational analysis and constrained optimization in finite-dimensional spaces. The main results concern developing extended calculus rules for these second-order constructions in both finite-dimensional and infinite-dimensional frameworks. We also provide new applications of partial … Read more

    Implementing cutting plane management and selection techniques

  • U. H. Suhl
  • Franz Wesselmann
    • Categories Cutting Plane Approaches Tags ,

    One main objective of research in the area of mixed-integer programming is developing cutting plane techniques to improve the solvability of mixed-integer programs (MIPs). Various cutting plane separators are typically available in MIP solvers. The large number of cutting planes generated by these separators, however, can pose a computational problem. Therefore, a sophisticated cut management … Read more