The convex hull property and topology of hypersurfaces with nonnegative curvature

We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure.     

求解随机凸规划概率约束问题的对偶算法

1引言随机规划中的概率约束问题在工程和管理中有广泛的应用.因为问题中包含非线性的概率约束,它们的求解非常困难.如果目标函数是线性的,问题的求解就比较容易.给出了一个求解随机线性规划概率约束问题的综述.原-对偶算法和切平面算法是比较有效的.在本文中,我们讨论随机凸规划概率约束问题:     

On the second conjugate of several convex functions in general normed vector spaces

When dealing with convex functions defined on a normed vector space X the biconjugate is usually considered with respect to the dual system (X, X ^*), that is, as a function defined on the initial space X. However, it is of interest to consider also the biconjugate as a function defined on the bidual X ^**. It is the aim of this note to calculate the biconjugate of the functions obtained by several operations which preserve convexity. In particular we recover the result of Fitzpatrick and Simons on the biconjugate of the maximum of two convex functions with a much simpler proof.      

Constructing the Solution to Robin's Equation

AMS (MOS): 45B05, 45C05

The standard method for obtaining a nontrivial solution to Robin's equation is by successive approximation. The usual proof of the convergence of the sequence obtained by this iteration procedure is based on Neumann's method of arithmetic mean, and requires that the surface over which the integral operator is defined be smooth and convex. Without assuming that the surface is convex, in this paper we give a new and simpler proof of the convergence of this sequence.     

On Solvability of Convex Noncoercive Quadratic Programming Problems

Using a known result on minimization of convex functionals on polyhedral cones, the Frank–Wolfe theorem, and basic linear algebra, we give a simple proof that the general convex quadratic programming problem which satisfies a natural necessary condition has a solution.     

The Monge problem in $ {\mathbb{R}^d} $: Variations on a theme

In a recent paper, the authors proved that, under natural assumptions on the first marginal, the Monge problem in \mathbbR^d {\mathbb{R}^d} for the cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of a uniformly convex norm, and we will also use very recent results by Ahmad, Kim, and McCann. This allows us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set in the particular case considered in this paper is original. Bibliography: 22 titles.     

Local asymptotic laws for the Brownian convex hull

Summary We prove a general theorem for the precise rate at which the convex hull of Brownian motion gets created. The latter result relates large deviation theory to P. Lévy's geometric proof of Strassen's law of the iterated logarithm. This also answers a question of S. Evans. Moreover, we give a partial solution to a question of J. Hammersley and P. Lévy regarding the slowness of the growth of the hull process. Several examples, some classical and some new, are given to illustrate the theorems. Finally, we present applications to the convex hull of random walks ind dimensions.     

An Eberlein–Šmulian type result for the weak* topology

We present a result on relative weak* compactness in the dual of a Banach space X that allows a short proof of both the Eberlein–Šmulian theorem and Šmulian’s characterisation of weak compactness of closed convex subsets of X.     

Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions

Computable lower and upper bounds on the optimal and dual optimal solutions of a nonlinear, convex separable program are obtained from its piecewise linear approximation. They provide traditional error and sensitivity measures and are shown to be attainable for some problems. In addition, the bounds on the solution can be used to develop an efficient solution approach for such programs, and the dual bounds enable us to determine a subdivision interval which insures the objective function accuracy of a prespecified level. A generalization of the bounds to certain separable, but nonconvex, programs is given and some numerical examples are included.     

Alternative proofs of the convergence properties of the conjugate-gradient method

For the problem of minimizing an unconstrained function, the conjugate-gradient method is shown to be convergent. If the function is uniformly strictly convex, the ultimate rate of convergence is shown to ben-step superlinear. If the Hessian matrix is Lipschitz continuous, the rate of convergence is shown to ben-step quadratic. All results are obtained for the reset version of the method and with a relaxed requirement on the solution of the stepsize problem. In addition to obtaining sharper results, the paper differs from previously published ones in the mode of proof which contains as a corollary the proof of finiteness of the conjugate-gradient method when applied to a quadratic problem rather than assuming that result.     

On the generic properties of convex optimization problems in conic form

We prove that strict complementarity, primal and dual nondegeneracy of optimal solutions of convex optimization problems in conic form are generic properties. In this paper, we say generic to mean that the set of data possessing the desired property (or properties) has strictly larger Hausdorff dimension than the set of data that does not. Our proof is elementary and it employs an important result due to Larman [7] on the boundary structure of convex bodies. Received: September 1997 / Accepted: May 2000?Published online November 17, 2000     

Characterizations for solidness of dual cones with applications

In this paper, some characterizations for the solidness of dual cones are established. As applications, we prove that a Banach space is reflexive if it contains a solid pointed closed convex cone having a weakly compact base, and prove an analogue of a Karamardian’s result for the linear complementarity problem in reflexive Banach spaces. The uniqueness of the solution of the linear complementarity problem is also discussed.     

THE ALEKSANDROV PROBLEM FOR UNIT DISTANCE PRESERVING MAPPING

1 IntroductionLet X and Y be two real metric spaces. A mappillg f: X ~ Y is called an isometryj ifd(f(x), f(y)) = d(x, y) for all x, y E X.A mapping f: X - Y satisfies the distance one preserving property (DOPP) if f for allx, y E X with d(x, y) = 1 it follows that d(f(x), f(y)) = 1.A mapping f: X ~ Y satisfies the strong distance one preserving property (SDOPP) ifffor all x, y E X with d(x, y) = 1 it follows that d(f(x), f(y)) = 1 and conversely.Problem(P) Let f: X - Y be a mappin…     

A connection between a convex programming problem and the LYM property on perfect graphs

We derive three equivalent conditions on a perfect graph concerning the optimal solution of a convex programming problem, the length-width inequality, and the simultaneous vertex covering by cliques and anticliques. By combining proof techniques including Lagrangian dual, Dilworth's Theorem, and Kuhn-Tucker Theorem, we establish a strong connection between the three topics. This provides new insights into the structure of perfect graphs. The famous Lubell-Yamamoto-Meschalkin (LYM) Property or Sperner Property for partially ordered sets is a specialization of our results to a subclass of perfect graphs.     

The least slope of a convex function and the maximal monotonicity of its subdifferential

In this paper, we give a direct proof of Rockafellar's result that the subdifferential of a proper convex lower-semicontinuous function on a Banach space is maximal monotone. Our proof is simpler than those that have appeared to date. In fact, we show that Rockafellar's result can be embedded in a more general situation in which we can quantify the degree of failure of monotonicity in terms of a quotient like the one that appears in the definition of Fréchet differentiability. Our analysis depends on the concepts of the least slope of a convex function, which is related to the steepest descent of optimization theory.The author would like to express his thanks to R. R. Phelps for reading a preliminary version of this paper and making some very valuable suggestions.     

A note on the paper “Fractional programming with convex quadratic forms and functions” by H.P. Benson

In this technical note, we give a short proof based on some standard results in convex analysis of some important characterization results listed in Theorems 3 and 4 of Benson [Benson, H.P., 2006. Fractional programming with convex quadratic forms and functions. European Journal of Operational Research]. Actually our result is slightly more general since we do not specify the nonempty convex set X. For clarity we use the same notation for the different equivalent optimization problems as done in Benson (2006).     

Minimum Distance to the Complement of a Convex Set: Duality Result

The subject of this paper is to study the problem of the minimum distance to the complement of a convex set. Nirenberg has stated a duality theorem treating the minimum norm problem for a convex set. We state a duality result which presents some analogy with the Nirenberg theorem, and we apply this result to polyhedral convex sets. First, we assume that the polyhedral set is expressed as the intersection of some finite collection of m given half-spaces. We show that a global solution is determined by solving m convex programs. If the polyhedral set is expressed as the convex hull of a given finite set of extreme points, we show that a global minimum for a polyhedral norm is obtained by solving a finite number of linear programs.     

Application of the alternating direction method of multipliers to separable convex programming problems

This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.     

Necessary and Sufficient Conditions for Solving Infinite-Dimensional Linear Inequalities

The existence of a feasible solution to a system of infinite-dimensional linear inequalities is characterized by a topological generalization of the Farkas Condition. If this result is specialized to a finite-dimensional vector space with finite positive cone, then a geometric proof of the classic Minkowski-Farkas Lemma is obtained. A dual version leads to an infinite-dimensional extension of the Theorem of the Alternative.     

A class of history-dependent variational-hemivariational inequalities

We consider a new class of variational-hemivariational inequalities which arise in the study of quasistatic models of contact. The novelty lies in the special structure of these inequalities, since each inequality of the class involve unilateral constraints, a history-dependent operator and two nondifferentiable functionals, of which at least one is convex. We prove an existence and uniqueness result of the solution. The proof is based on arguments on elliptic variational-hemivariational inequalities obtained in our previous work [23], combined with a fixed point result obtained in [30]. Then, we prove a convergence result which shows the continuous dependence of the solution with respect to the data. Finally, we present a quasistatic frictionless problem for viscoelastic materials in which the contact is modeled with normal compliance and finite penetration and the elasticity operator is associated to a history-dependent Von Mises convex. We prove that the variational formulation of the problem cast in the abstract setting of history-dependent quasivariational inequalities, with a convenient choice of spaces and operators. Then we apply our general results in order to prove the unique weak solvability of the contact problem and its continuous dependence on the data.