Let F be a subset of the n-dimensional Euclidean space R n represented in terms of a compact convex subset C 0 and a set P F of nitely or innnitely many quadratic functions on R n such that F = fx 2 C 0 : p(x) 0 (8p() 2 P F)g. In this paper, we investigate some fundamental properties related to the nite convergence of the successive SDP (semideenite programming) relaxation method proposed by th...

We introduce a notion of quasi regularity for points with respect to the inclusion F (x) ∈ C, where F is a nonlinear Fréchet differentiable function from Rv to Rm. When C is the set of minimum points of a convex real-valued function h on Rm and F ′ satisfies the L-average Lipschitz condition of Wang, we use the majorizing function technique to establish the semilocal linear/quadratic convergenc...

Let Mφ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f : X → R, a measure μ ∈ Mφ is called f -maximizing if ∫ f dμ = sup{ ∫ f dm : m ∈Mφ}. It is shown that if μ is any ergodic measure in Mφ, then there exists a continuous function whose unique maximizing measure is μ. More generally, if E is a non-e...

For fixed ƒ and r this mean modulus Mt(r;f) as a function of / is continuous, nonnegative, nondecreasing, and is bounded above by the maximum modulus of ƒ on C(r) [l, 2 ] . 1 Therefore the limit of Mt(r;f) exists as /—>0 and /—» oo. This limit is defined to be the mean modulus of ƒ on C(r) of order 0 and of order oo respectively. I t may be shown that the mean modulus of order 0 is the geometri...

Of key importance in convex analysis and optimization is the notion of duality, and in particular that of Fenchel duality. This work explores improvements to existing algorithms for the symbolic calculation of subdifferentials and Fenchel conjugates of convex functions defined on the real line. More importantly, these algorithms are extended to enable the symbolic calculation of Fenchel conjuga...

In this note we first redefine the notion of a fuzzy hypervectorspace (see [1]) and then introduce some further concepts of fuzzy hypervectorspaces, such as fuzzy convex and balance fuzzy subsets in fuzzy hypervectorspaces over valued fields. Finally, we briefly discuss on the convex (balanced)hull of a given fuzzy set of a hypervector space.

It is known that one can fold a convex polyhedron from a non-overlapping face unfolding, but the complexity of the algorithm in [MP] remains an open problem. In this paper we show that every convex polyhedron P ⊂ R can be obtained in polynomial time, by starting with a cube which contains P and sequentially cutting out the extra parts of the surface. Our main tool is of independent interest. We...

We consider the convex optimization problem minx{f(x) : gj(x) ≤ 0, j = 1, . . . , m} where f is convex, the feasible set K is convex and Slater’s condition holds, but the functions gj ’s are not necessarily convex. We show that for any representation of K that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minim...

in this paper, we first present a new important property for bouligand tangent cone (contingent cone) of a star-shaped set. we then establish optimality conditions for pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.