Generalized Convexity/Monoticity

Optimality conditions of set-valued optimization problem involving relative algebraic interior in ordered linear spaces

  • Peng Jian-Wen
  • Xinmin Yang
  • Zhou Zhi-Ang
    • Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization Tags

    In this paper, firstly, a generalized subconvexlike set-valued map involving the relative algebraic interior is introduced in ordered linear spaces. Secondly, some properties of a generalized subconvexlike set-valued map are investigated. Finally, the optimality conditions of set-valued optimization problem are established. Citation {\bf AMS 2010 Subject Classifications:} 90C26, 90C29, 90C30ArticleDownload View PDF

    Identifying Active Manifolds in Regularization Problems

  • Warren Hare
    • Categories Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , ,

    In this work we consider the problem $\min_x \{ f(x) + P(x) \}$, where $f$ is $\mathcal{C}^2$ and $P$ is nonsmooth, but contains an underlying smooth substructure. Specifically, we assume the function $P$ is prox-regular partly smooth with respect to a active manifold $\M$. Recent work by Tseng and Yun \cite{tseng-yun-2009}, showed that such a … Read more

    Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions

  • X. X. Huang
  • J. Zhang
    • Categories Generalized Convexity/Monoticity, Nonlinear Optimization Tags , , , ,

    In this paper, we apply the quadratic penalization technique to derive strong Lagrangian duality property for an inequality constrained invex program. Our results extend and improve the corresponding results in the literature. CitationBazara, M. S. and Shetty, C. M., Nonlinear Programming Theory and Algorithms, John Wiley \& Sons, New York, 1979. Ben-Israel, A. and Mond, … Read more

    A Redistributed Proximal Bundle Method for Nonconvex Optimization

  • Warren Hare
  • Claudia Sagastizábal
    • Categories Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , , ,

    Proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives. This naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting. This work proposes an approach based on generating cutting-planes models, not of the objective function … Read more

    Nonsmooth Optimization for Production Theory

  • Yoshihiro Tanaka
    • Categories Generalized Convexity/Monoticity

    Production theory needs generalizations so that it can incorporate broader class of production functions. A generalized Hotelling’s lemma and a generalized Shephard’s lemma in economic theory, which are established in virtue of nonsmooth analysis under the assumption of upper semicontinuity on production functions. Continuity of factor inputs with respect to a change of the factor … Read more

    Monotonicity of L”{o}wner Operators and Its Applications to Symmetric Cone Complementarity Problems

  • Lingchen Kong
  • Levent Tunçel
  • Naihua Xiu
    • Categories Complementarity and Variational Inequalities, Generalized Convexity/Monoticity, Semi-definite Programming Tags , , , ,

    This paper focuses on monotone L\”{o}wner operators in Euclidean Jordan algebras and their applications to the symmetric cone complementarity problem (SCCP). We prove necessary and sufficient conditions for locally Lipschitz L\”{o}wner operators to be monotone, strictly monotone and strongly monotone. We also study the relationship between monotonicity and operator-monotonicity of L\”{o}wner operators. As a by-product … Read more

    Nonsmooth Quasiconcave Programming

  • Yoshihiro Tanaka
    • Categories Generalized Convexity/Monoticity Tags , , ,

    This paper is devoted to optimality conditions for nonsmooth quasiconcave programming. Arrow and Enthoven (1961) formulate several economic problems into quasiconcave programming, and give a sufficient condition for smooth quasiconcave programming in their epoch-making and comprehensive paper. In this paper, generalized necessary and sufficient conditions for nonsmooth quasiconcave programming have been derived in terms of … Read more

    A Proximal Point Algorithm with Bregman Distances for Quasiconvex Optimization over the Positive Orthant

  • João Xavier da Cruz Neto
  • Paulo Roberto Oliveira
  • Antoine Soubeyran
  • Sissy da S. Souza
    • Categories Generalized Convexity/Monoticity Tags , , ,

    We present an interior proximal point method with Bregman distance, whose Bregman function is separable and the zone is the interior of the positive orthant, for solving the quasiconvex optimization problem under nonnegative constraints. We establish the well-definedness of the sequence generated by our algorithm and we prove convergence to a solution point when the … Read more

    Simplex-type algorithm for optimizing a pseudolinear quadratic fractional function over a polytope

  • Miklós Ujvári
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity

    Recently Cambini and Carosi described a characterization of pseudolinearity of quadratic fractional functions. A reformulation of their result was given by Rapcsák. Using this reformulation, in this paper we describe an alternative proof of the Cambini–Carosi Theorem. Our proof is shorter than the proof given by Cambini–Carosi and less involved than the proof given by … Read more

    A Proximal Point Algorithm with phi-Divergence to Quasiconvex Programming

  • F. G. M. Cunha
    • Categories Generalized Convexity/Monoticity, Nonlinear Optimization Tags , ,

    We use the proximal point method with the phi-divergence given by phi(t) = t – log t – 1 for the minimization of quasiconvex functions subject to nonnegativity constraints. We establish that the sequence generated by our algorithm is well-defined in the sense that it exists and it is not cyclical. Without any assumption of … Read more