Minimizing Increasing Star-shaped Functions Based on Abstract Convexity

We study a broad class of increasing non-convex functions whose level sets are star shaped with respect to infinity. We show that these functions (we call them ISSI functions) are abstract convex with respect to the set of min-type functions and exploit this fact for their minimization. An algorithm is proposed for solving global optimization problems with an ISSI objective function and its numerical performance is discussed.     

Efficient and weakly efficient solutions for the vector topical optimization

In this paper, we consider some scalarization functions, which consist of the generalized min-type function, the so-called plus-Minkowski function and their convex combinations. We investigate the abstract convexity properties of these scalarization functions and use them to identify the maximal points of a set in an ordered vector space. Then, we establish some versions of Farkas type results for the infinite inequality system involving vector topical functions. As applications, we obtain the necessary and sufficient conditions of efficient solutions and weakly efficient solutions for a vector topical optimization problem, respectively.     

Farkas-type theorems for positively homogeneous systems in ordered topological vector spaces

In this paper, we present versions of the Farkas Lemma and the Gale Lemma for a semi-infinite system involving positively homogeneous functions in a topological vector space. In particular, we present two such versions for a semi-infinite system containing min-type functions. Our main theoretical tool is abstract convexity.     

Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions, we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann–Dirichlet boundary conditions and more general curvature-dependent speeds.     

Further Applications of the Additive Min-type Coupling Function

Applying for the additive min-type coupling function $$ \varphi (x, w) = \mathop {\min }\limits_{1 \le i \le n} (x_1 + w_i )\quad (x = (x_i ), w = (w_i ) \in {\shadR}^n ) $$ the techniques from abstract convex analysis, developed in [I. Singer (1997). Abstract Convex Analysis . Wiley-Interscience, New York], we show that the conjugate of type Lau f L ( } ) of f with respect to } has the simple expression $ f^{L(\varphi )} (w) = - f( - w)\, (w \in {\shadR}^n ) $ if and only if f is increasing and upper semi-continuous. As a consequence, we obtain that for topical (i.e., increasing additively homogeneous) functions f the Fenchel-Moreau conjugate of f with respect to } and the conjugate of type Lau of f with respect to } coincide.     

Global existence and blow-up for harmonic map heat flow

For degree-one equivariant maps on bounded domains, the question of finite-time blow-up vs. global existence of solutions to the harmonic map heat flow has been well studied. In this paper we study the Cauchy problem for degree-m equivariant harmonic map heat flow from (2+1)-dimensional space-time into the 2-sphere with initial energy close to the energy of harmonic maps. It is proved that solutions are globally smooth for m?4, whereas for m=1, we show that finite-time singularities can form for this class of data.     

A new class of complementarity functions for symmetric cone complementarity problems

A popular approach to solving the complementarity problem is to reformulate it as an equivalent equation system via a complementarity function. In this paper, we propose a new class of functions, which contains the penalized natural residual function and the penalized Fischer–Burmeister function for symmetric cone complementarity problems. We show that this class of functions is indeed a class of complementarity functions. We finally prove that the merit function of this new class of complementarity functions is coercive.     

On the local representation of piecewise smooth equations as a Lipschitz manifold

We study systems of equations, F(x)=0$F(x)=0$, given by piecewise differentiable functions F:Rⁿ→R^k$F:{\mathbb{R}}^n\to {\mathbb{R}}^k$, k?n$k?n$. The focus is on the representability of the solution set locally as an (n−k)$(n-k)$-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarke?s generalized Jacobians. The SFRA claims the existence of a basis in which Clarke?s inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarke?s generalized Jacobian. The article is devoted to the conjectured equivalence of SFRA and FRA. For min-type functions, we give reformulations of SFRA and FRA using orthogonal projections, basis enlargements, cross products, dual variables, as well as via exponentially many convex cones. The equivalence of SFRA and FRA is shown to be true for min-type functions in the new case k=3$k=3$.     

Function Decompositions Related to the Luzin N-property

We introduce a class of continuous completely regular functions satisfying the N-property. We obtain a decomposition of an arbitrary continuous function into the sum of two functions the first of which is completely regular and the second does not enjoy the N-property. We define a class of strongly regular Borel functions for which we prove the Luzin N-property. We demonstrate that the image of every Lebesgue measurable set of a strongly regular function is measurable. From an arbitrary Borel function we extract a strongly regular function and a function that does not enjoy the N-property.     

On merit functions for <Emphasis Type="Italic">p</Emphasis>-order cone complementarity problem

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.     

A new class of smoothing complementarity functions over symmetric cones

A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Fréchet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones.     

The arity gap of order-preserving functions and extensions of pseudo-Boolean functions

The aim of this paper is to classify order-preserving functions according to their arity gap. Noteworthy examples of order-preserving functions are the so-called aggregation functions. We first explicitly classify the Lovász extensions of pseudo-Boolean functions according to their arity gap. Then we consider the class of order-preserving functions between partially ordered sets, and establish a similar explicit classification for this function class.     

A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

In this paper, we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite. We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function.     

A New Class of Complementarity Function and the Boundedness of Its Merit Function for Symmetric Cone Complementarity Problem

In this paper, we introduce a new class of two-parametric penalized function, which includes the penalized minimum function and the penalized Fischer-Burmeister flmc- tion over symmetric cone complementarity problems. We propose that this class of function is a class of complementarity functions（C-function）. Moreover, its merit function has bounded level set under a weak condition.     

Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions

In the geometries of stratified groups, we show that H-convex functions locally bounded from above are locally Lipschitz continuous and that the class of v-convex functions exactly corresponds to the class of upper semicontinuous H-convex functions. As a consequence, v-convex functions are locally Lipschitz continuous in every stratified group. In the class of step 2 groups we characterize locally Lipschitz H-convex functions as measures whose distributional horizontal Hessian is positive semidefinite. In Euclidean space the same results were obtained by Dudley and Reshetnyak. We prove that a continuous H-convex function is a.e. twice differentiable whenever its second order horizontal derivatives are Radon measures.     

Lipschitz continuity of the gradient of a one-parametric class of SOC merit functions

In this article, we show that a one-parametric class of SOC merit functions has a Lipschitz continuous gradient; and moreover, the Lipschitz constant is related to the parameter in this class of SOC merit functions. This fact will lay a building block when the merit function approach as well as the Newton-type method are employed for solving the second-order cone complementarity problem with this class of merit functions.     

On a new class of function systems of Faber-Schauder type

In this paper, we introduce a new class of function systems generalizing the classical Faber-Schauder system. Under the condition that the generating sequence is bounded, we show that systems of such a class constitute bases in the space of continuous functions and prove some properties of series expansions of functions in these systems.