Examples of r-convex functions and characterizations of r-convex functions associated with second-order cone (original) (raw)

The convex and monotone functions associated with second-order cone

2006

Like the matrix-valued functions used in solutions methods for semidefinite programs (SDPs) and semidefinite complementarity problems (SDCPs), the vector-valued functions associated with second-order cones are defined analogously and also used in solutions methods for second-order-cone programs (SOCPs) and second-order-cone complementarity problems (SOCCPs). In this article, we study further about these vector-valued functions associated with second-order cones (SOCs). In particular, we define the so-called SOC-convex and SOC-monotone functions for any given function f : R ! R. We discuss the SOC-convexity and SOC-monotonicity for some simple functions, e.g., fðtÞ ¼ t 2 , t 3 , 1=t, t 1=2 , jtj, and [t] þ. Some characterizations of SOC-convex and SOC-monotone functions are studied, and some conjectures about the relationship between SOC-convex and SOC-monotone functions are proposed.

Scalar characterizations of weakly cone-convex and weakly cone-quasiconvex functions

Nonlinear Analysis: Theory, Methods & Applications, 2010

The principal aim of this paper is to show that weakly cone-convex vector-valued functions, as well as weakly cone-quasiconvex vector-valued functions, can be characterized in terms of usual weakly convexity and weakly quasiconvexity of certain real-valued functions, defined by means of the extreme directions of the polar cone or by Gerstewitz's scalarization functions.

Generalized (Phi, Rho)-convexity in nonsmooth vector optimization over cones

An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 2016

In this paper, new classes of cone-generalized (Phi,Rho)-convex functions are introduced for a nonsmooth vector optimization problem over cones, which subsume several known studied classes. Using these generalized functions, various sufficient Karush-Kuhn-Tucker (KKT) type nonsmooth optimality conditions are established wherein Clarke's generalized gradient is used. Further, we prove duality results for both Wolfe and Mond-Weir type duals under various types of cone-generalized (Phi,Rho)-convexity assumptions.Phi,Rho

Generalized higher-order cone-convex functions and higher-order duality in vector optimization

Annals of Operations Research, 2017

In this paper, we introduce a new class of higher-order cone-convex, (K 1 , K 2)pseudoconvex and quasiconvex functions which encapsulates several already known functions. Higher-order sufficient optimality conditions have been established for a vector optimization problem over cones by using these functions, under weaker conditions on multipliers as compared to other papers in this domain. Wolfe type and Mond-Weir type higher-order duals are formulated and corresponding duality results are established. A number of previously studied problems appear as special cases of our primal-dual models. In case of nonlinear programming problem, our higher-order duals reduce to the corresponding higher-order duals given by Mangasarian (

Quasi-convex functions and applications to optimality conditions in nonlinear programming

Applied Mathematics Letters, 2001

Communicated by F. Clarke Abstract-We establish necessary and sufficient optimality conditions for quasi-convex programming. First, we treat some properties of the normal cone to the level set of a lower semicontinuous quasi-convex function defined on a Banach space. Next, we get our conditions via some variational inequalities. (~

Second-order characterization of convex functions and its applications

Journal of Applied Analysis, 2019

Some developments of the second-order characterizations of convex functions are investigated by using the coderivative of the subdifferential mapping. Furthermore, some applications of the second-order subdifferentials in optimization problems are studied.

Circular cone convexity and some inequalities associated with circular cones

The study of this paper consists of two aspects. One is characterizing the so-called circular cone convexity of f by exploiting the second-order differentiability of f L θ ; the other is introducing the concepts of determinant and trace associated with circular cone and establishing their basic inequalities. These results show the essential role played by the angle θ , which gives us a new insight when looking into properties about circular cone. MSC: 26A27; 26B05; 26B35; 49J52; 90C33; 65K05

Second-order optimality and duality in vector optimization over cones

Statistics, Optimization & Information Computing, 2016

In this paper, we introduce the notion of a second-order cone-convex function involving second-order directional derivative. Also, second-order cone-pseudoconvex, second-order cone-quasiconvex and other related functions are defined. Second-order optimality and Mond-Weir type duality results are derived for a vector optimization problem over cones using the introduced classes of functions.