Convex and Nonsmooth Optimization

Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems

  • Xuan Duc Ha Truong
    • Categories Complementarity and Variational Inequalities, Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , , , , , , , , , , ,

    We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in … Read more

    A Practical Relative Error Criterion for Augmented Lagrangians

  • Jonathan Eckstein
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , ,

    This paper develops a new error criterion for the approximate minimization of augmented Lagrangian subproblems. This criterion is practical in the sense that it requires only information that is ordinarily readily available, such as the gradient (or a subgradient) of the augmented Lagrangian. It is also “relative” in the sense of relative error criteria for … Read more

    Complexity of variants of Tseng’s modified F-B splitting and Korpelevich’s methods for generalized variational inequalities with applications to saddle point and convex optimization problems

  • Renato D.C. Monteiro
  • Benar Fux Svaiter
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Convex Optimization Tags , , , , ,

    In this paper, we consider both a variant of Tseng’s modified forward-backward splitting method and an extension of Korpelevich’s method for solving generalized variational inequalities with Lipschitz continuous operators. By showing that these methods are special cases of the hybrid proximal extragradient (HPE) method introduced by Solodov and Svaiter, we derive iteration-complexity bounds for them … Read more

    Optimal Stochastic Approximation Algorithms for Strongly Convex Stochastic Composite Optimization I: a Generic Algorithmic Framework

  • Saeed Ghadimi
  • Guanghui Lan
    • Categories Convex Optimization, Data-Mining, Stochastic Programming Tags , , , , ,

    In this paper we present a generic algorithmic framework, namely, the accelerated stochastic approximation (AC-SA) algorithm, for solving strongly convex stochastic composite optimization (SCO) problems. While the classical stochastic approximation (SA) algorithms are asymptotically optimal for solving differentiable and strongly convex problems, the AC-SA algorithm, when employed with proper stepsize policies, can achieve optimal or … Read more

    Feasible and accurate algorithms for covering semidefinite programs

  • Garud Iyengar
  • Cliff Stein
  • David Phillips
    • Categories Approximation Algorithms, Nonsmooth Optimization, Semi-definite Programming Tags , , ,

    In this paper we describe an algorithm to approximately solve a class of semidefinite programs called covering semidefinite programs. This class includes many semidefinite programs that arise in the context of developing algorithms for important optimization problems such as sparsest cut, wireless multicasting, and pattern classification. We give algorithms for covering SDPs whose dependence on … Read more

    Optimality Conditions and Duality for Nonsmooth Multiobjective Optimization Problems with Cone Constraints and Applications

  • Jiawei Chen
  • Zhong-Ping Wan
  • Zheng Yue
    • Categories Nonsmooth Optimization, Other Topics Tags , , , , , ,

    In this work, a nonsmooth multiobjective optimization problem involving generalized invexity with cone constraints and Applications (for short, (MOP)) is considered. The Kuhn-Tucker necessary and sufficient conditions for (MOP) are established by using a generalized alternative theorem of Craven and Yang. The relationship between weakly efficient solutions of (MOP) and vector valued saddle points of … Read more

    A splitting method for separate convex programming with linking linear constraints

  • Bingsheng He
  • Tao Min
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , , ,

    We consider the separate convex programming problem with linking linear constraints, where the objective function is in the form of the sum of m individual functions without crossed variables. The special case with m=2 has been well studied in the literature and some algorithms are very influential, e.g. the alternating direction method. The research for … Read more

    Alternating proximal algorithms for constrained variational inequalities. Application to domain decomposition for PDE’s

  • Hédy Attouch
  • A Cabot
  • Pierre Frankel
  • Juan Peypouquet
    • Categories Convex Optimization Tags , , , , ,

    Let $\cX,\cY,\cZ$ be real Hilbert spaces, let $f : \cX \rightarrow \R\cup\{+\infty\}$, $g : \cY \rightarrow \R\cup\{+\infty\}$ be closed convex functions and let $A : \cX \rightarrow \cZ$, $B : \cY \rightarrow \cZ$ be linear continuous operators. Let us consider the constrained minimization problem $$ \min\{f(x)+g(y):\quad Ax=By\}.\leqno (\cP)$$ Given a sequence $(\gamma_n)$ which tends toward … Read more

    Implicit Multifunction Theorems in complete metric spaces

  • Van Ngai Huynh
  • Huu Tron Nguyen
  • Michel Thera
    • Categories Infinite Dimensional Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper, we establish some new characterizations of the metric regularity of implicit multifunctions in complete metric spaces by using the lower semicontinuous envelopes of the distance functions for set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the … Read more