Generalized Convexity/Monoticity

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

    An Extension of the Proximal Point Method for Quasiconvex Minimization

  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
    • Categories Generalized Convexity/Monoticity, Unconstrained Optimization Tags , , ,

    In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on the Euclidean space and the nonnegative orthant. For the unconstrained minimization problem, assumming that the function is bounded from below and lower semicontinuous we prove that iterations fxkg given by 0 2 b@(f(:)+(k=2)jj:􀀀xk􀀀1jj2)(xk) are … Read more

    Steepest descent method for quasiconvex minimization on Riemannian manifolds

  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
  • E. M. Quispe
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity

    This paper extends the full convergence of the steepest descent algorithm with a generalized Armijo search and a proximal regularization to solve quasiconvex minimization problems defined on complete Riemannian manifolds. Previous convergence results are obtained as particular cases of our approach and some examples in non Euclidian spaces are given. Citation J. Math. Anal. Appl. … Read more

    Some remarks about the transformation of Charnes and Cooper

  • Ezio Marchi
    • Categories Applications - OR and Management Sciences, Generalized Convexity/Monoticity Tags , , ,

    In this paper we extend in a simple way the transformation of Charnes and Cooper to the case where the functional ratio to be considered are of similar polynomial Citation Universidad de San Luis Ejercito de Los Andes 950 San Luis(5700) Argentina Article Download View Some remarks about the transformation of Charnes and Cooper

    Proximal Point Methods for Quasiconvex and Convex Functions With Bregman Distances

  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
    • Categories Convex Optimization, Generalized Convexity/Monoticity Tags ,

    This paper generalizes the proximal point method using Bregman distances to solve convex and quasiconvex optimization problems on noncompact Hadamard manifolds. We will proved that the sequence generated by our method is well defined and converges to an optimal solution of the problem. Also, we obtain the same convergence properties for the classical proximal method, … Read more

    A Proximal Method for Identifying Active Manifolds

  • Warren Hare
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity Tags , , ,

    The minimization of an objective function over a constraint set can often be simplified if the “active manifold” of the constraints set can be correctly identified. In this work we present a simple subproblem, which can be used inside of any (convergent) optimization algorithm, that will identify the active manifold of a “prox-regular partly smooth” … Read more