convex analysis

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

  • Yun-Bin Zhao
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: \emph{When is the product of finitely many positive … Read more

    The Legendre-Fenchel Conjugate of the Product of Two positive definite Quadratic Forms

  • Yun-Bin Zhao
    • Categories Convex Optimization Tags , , ,

    It is well-known that the Legendre-Fenchel conjugate of a positive definite quadratic form can be explicitly expressed as another positive definite quadratic form, and that the conjugate of the sum of several positive definite quadratic forms can be expressed via inf-convolution. However, the Legendre-Fenchel conjugate of the product of two positive definite quadratic forms is … Read more

    Semi-infinite programming, duality, discretization and optimality conditions

  • Alexander Shapiro
    • Categories Semi-infinite Programming Tags , , , , ,

    The aim of this paper is to give a survey of some basic theory of semi-infinite programming. In particular, we discuss various approaches to derivations of duality, discretization and first and second order optimality conditions. Some of the surveyed results are well known while others seem to be less noticed in that area of research. … Read more

    On Non-Convex Quadratic Programming with Box Constraints

  • Samuel Burer
  • Adam N. Letchford
    • Categories 0-1 Programming, Convex Optimization, Global Optimization Theory Tags , ,

    Non-Convex Quadratic Programming with Box Constraints is a fundamental NP-hard global optimisation problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterise their extreme points and vertices, show their invariance under certain affine transformations, … Read more

    Graph Implementations for Nonsmooth Convex Programs

  • Stephen Boyd
  • Michael Grant
    • Categories Convex Optimization, Modeling Languages and Systems, Nonsmooth Optimization Tags , ,

    We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved, using interior-point methods for smooth or cone convex programs. CitationTo … Read more

    Conditional Risk Mappings

  • Andrzej Ruszczyński
  • Alexander Shapiro
    • Categories Finance and Economics, Stochastic Programming Tags , , , , ,

    We introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties. In particular, we prove a representation theorem for conditional risk mappings in terms of conditional expectations. We also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings. CitationPreprintArticleDownload View PDF

    Optimization of Convex Risk Functions

  • Andrzej Ruszczyński
  • Alexander Shapiro
    • Categories Finance and Economics, Stochastic Programming Tags , , , ,

    We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions. CitationPreprintArticleDownload View PDF