An example of bad convex function

An example of a convex function having the gradient at each point (x, 0, ..., 0),x>0, which does not converge, whenx tends to zero, is given.The author would like to thank Professor Giannessi for his help, which led to an improved version of this paper.     

Approximate first-order and second-order directional derivatives of a marginal function in convex optimization

Given a convex functionf: ^p × ^q (–, +], the marginal function is defined on ^p by (x)=inf{f(x, y)|y ^q }. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of atx ₀ in terms of those off at (x ₀,y ₀), wherey ₀ is any element for which (x ₀)=f(x ₀,y ₀).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.     

Minimization of convex functions on the convex hull of a point set

A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in R ⁿ is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal to m. The method is based on a new procedure for finding the projection of the gradient of the objective function onto a simplicial cone in R ^m , which is the tangent cone at the current point to the simplex defined by the usual constraints on barycentric coordinates. It is shown that this projection can be computed in O(m log m) operations. For strictly convex quadratic functions, the basic method can be refined to a noniterative method terminating with the optimal solution.     

Calculating the cone of directions of constancy

This note presents an algorithm that finds the cone of directions of constancy of a differentiable, faithfully convex function.This work was supported by the National Research Council of Canada. The author is indebted to Professor S. Zlobec for suggesting the topic and for his guidance.     

The gap function of a convex program

The gap function expresses the duality gap of a convex program as a function of the primal variables only. Differentiability and convexity properties are derived, and a convergent minimization algorithm is given. An example gives a simple one-variable interpretation of weak and strong duality. Application to user-equilibrium traffic assignment yields an appealing alternative optimization problem.     

The conjugate of the difference of convex functions

Given an arbitrary functiong and a convex functionh, we derive the expression of the conjugate ofg —h via a simple proof.     

A general theory of almost convex functions

Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -almost convex iff for all and the inequality

holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function is constructed with the properties that it vanishes on the vertices of and if is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

     

一类二层凸规划的分解法

研究了一类二层凸规划和与之相应的凸规划问题的等价性．并讨论了这类凸规划的对偶性和鞍点问题,最后给出了求解这类二层凸规划的一个分解法．     

Penalty function theory for general convex programming problems

A class of penalty functions for solving convex programming problems with general constraint sets is considered. Convergence theorems for penalty methods are established by utilizing the concept of infimal convergence of a sequence of functions. It is shown that most existing penalty functions are included in our class of penalty functions.     

Stability of linearly constrained convex quadratic programs

This paper establishes a simple necessary and sufficient condition for the stability of a linearly constrained convex quadratic program under perturbations of the linear part of the data, including the constraint matrix. It also establishes results on the continuity and differentiability of the optimal objective value of the program as a function of a parameter specifying the magnitude of the perturbation. The results established herein directly generalize well-known results on the stability of linear programs.     

An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes

In this paper, we constructively derive an explicit characterization of the convex envelope of a bilinear function over a special type of polytope in ². Our motivation stems from the use of such functions for deriving strengthened lower bounds within the context of a branch-and-bound algorithm for solving bilinear programming problems. For the case of polytopes with no edges having finite positive slopes, that is polytopes with downward sloping edges (which we call D-polytopes), we obtain a direct, explicit characterization of the convex envelope. This case subsumes the analysis of Al-Khayyal and Falk (1983) for constructing the convex envelope of a bilinear function over a rectangle in ². For non-D-polytopes, the analysis is more complex. We propose three strategies for this case based on (i) encasing the region in a D-polytope, (ii) employing a discretization technique, and (iii) providing an explicit characterization over a triangle along with a triangular decomposition approach. The analysis is illustrated using numerical examples.     

Existence and sum decomposition of vertex polyhedral convex envelopes

Convex envelopes are a very useful tool in global optimization. However finding the exact convex envelope of a function is a difficult task in general. This task becomes considerably simpler in the case where the domain is a polyhedron and the convex envelope is vertex polyhedral, i.e., has a polyhedral epigraph whose vertices correspond to the vertices of the domain. A further simplification is possible when the convex envelope is sum decomposable, i.e., the convex envelope of a sum of functions coincides with the sum of the convex envelopes of the summands. In this paper we provide characterizations and sufficient conditions for the existence of a vertex polyhedral convex envelope. Our results extend and unify several results previously obtained for special cases of this problem. We then characterize sum decomposability of vertex polyhedral convex envelopes, and we show, among else, that the vertex polyhedral convex envelope of a sum of functions coincides with the sum of the vertex polyhedral convex envelopes of the summands if and only if the latter sum is vertex polyhedral.     

A conic quadratic formulation for a class of convex congestion functions in network flow problems

In this paper we consider a multicommodity network flow problem with flow routing and discrete capacity expansion decisions. The problem involves trading off congestion and capacity assignment (or expansion) costs. In particular, we consider congestion costs involving convex, increasing power functions of flows on the arcs. We first observe that under certain conditions the congestion cost can be formulated as a convex function of the capacity level and the flow. Then, we show that the problem can be efficiently formulated by using conic quadratic inequalities. As most of the research on this problem is devoted to heuristic approaches, this study differs in showing that the problem can be solved to optimum by branch-and-bound solvers implementing the second-order cone programming (SOCP) algorithms. Computational experiments on the test problems from the literature show that the continuous relaxation of the formulation gives a tight lower bound and leads to optimal or near optimal integer solutions within reasonable CPU times.     

Imputing a variational inequality function or a convex objective function: A robust approach

To impute the function of a variational inequality and the objective of a convex optimization problem from observations of (nearly) optimal decisions, previous approaches constructed inverse programming methods based on solving a convex optimization problem [17], [7]. However, we show that, in addition to requiring complete observations, these approaches are not robust to measurement errors, while in many applications, the outputs of decision processes are noisy and only partially observable from, e.g., limitations in the sensing infrastructure. To deal with noisy and missing data, we formulate our inverse problem as the minimization of a weighted sum of two objectives: 1) a duality gap or Karush–Kuhn–Tucker (KKT) residual, and 2) a distance from the observations robust to measurement errors. In addition, we show that our method encompasses previous ones by generating a sequence of Pareto optimal points (with respect to the two objectives) converging to an optimal solution of previous formulations. To compare duality gaps and KKT residuals, we also derive new sub-optimality results defined by KKT residuals. Finally, an implementation framework is proposed with applications to delay function inference on the road network of Los Angeles, and consumer utility estimation in oligopolies.     

On perimeters of sections of convex polytopes

In his book “Geometric Tomography” Richard Gardner asks the following question. Let P and Q be origin-symmetric convex bodies in R³ whose sections by any plane through the origin have equal perimeters. Is it true that P=Q? We show that the answer is “Yes” in the class of origin-symmetric convex polytopes. The problem is treated in the general case of Rn.     

A simplified test for optimality

A simplification of recent characterizations of optimality in convex programming involving the cones of decrease and constancy of the objective and constraint functions is presented. In the original characterization due to Ben-Israelet al., optimality was verified or a feasible direction of decrease was determined by considering a number of sets equal to the number of subsets of the set of binding constraints. By first finding the set of constraints which is binding at every feasible point, it is possible to verify optimality or determine a feasible direction of decrease by considering a single set. In the case of faithfully convex functions, this set can be found by solving at mostp systems of linear equations and inequalities, wherep is the number of constraints.This work was partly supported by NSF Grant No. Eng 76-10260.     

On duality for square root convex programs

Conjugate function theory is used to develop dual programs for nonseparable convex programs involving the square root function. This function arises naturally in finance when one measures the risk of a portfolio by its variance–covariance matrix, in stochastic programming under chance constraints and in location theory.     

关于一些凸规划问题的复杂性研究结果

目前,已发表了大量研究各类不同凸规划的低复杂度的障碍函数方法的文章. 利用自和谐理论,对不同的几类凸规划问题构造相应的对数障碍函数,通过两个引理证明这些凸规划问题相应的对数障碍函数都满足自和谐,根据Nesterov 和Nemirovsky的工作证明了所给问题的内点算法具有多项式复杂性.     

指数凸函数的积分不等式及其在Gamma函数中的应用

仿对数凸函数的概念,给出指数凸函数的定义,并证明有关指数凸函数的几个积分不等式,作为应用,得到一个新的Kershaw型双向不等式.