A proximal point algorithm revisit on the alternating direction method of multipliers

The alternating direction method of multipliers(ADMM)is a benchmark for solving convex programming problems with separable objective functions and linear constraints.In the literature it has been illustrated as an application of the proximal point algorithm(PPA)to the dual problem of the model under consideration.This paper shows that ADMM can also be regarded as an application of PPA to the primal model with a customized choice of the proximal parameter.This primal illustration of ADMM is thus complemental to its dual illustration in the literature.This PPA revisit on ADMM from the primal perspective also enables us to recover the generalized ADMM proposed by Eckstein and Bertsekas easily.A worst-case O(1/t)convergence rate in ergodic sense is established for a slight extension of Eckstein and Bertsekas’s generalized ADMM.     

广义几何规划的压缩共轭梯度路径非单调信赖域算法(英文)

In this paper,on the basis of making full use of the characteristics of unconstrained generalized geometric programming（GGP）,we establish a nonmonotonic trust region algorithm via the conjugate path for solving unconstrained GGP problem.A new type of condensation problem is presented,then a particular conjugate path is constructed for the problem,along which we get the approximate solution of the problem by nonmonotonic trust region algorithm,and further prove that the algorithm has global convergence and quadratic convergence properties.     

Global optimization method for linear multiplicative programming

In this paper, a new global algorithm is presented to globally solve the linear multiplicative programming(LMP). The problem(LMP) is firstly converted into an equivalent programming problem(LMP(H))by introducing p auxiliary variables. Then by exploiting structure of(LMP(H)), a linear relaxation programming(LP(H)) of(LMP(H)) is obtained with a problem(LMP) reduced to a sequence of linear programming problems. The algorithm is used to compute the lower bounds called the branch and bound search by solving linear relaxation programming problems(LP(H)). The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Some examples are given to illustrate the feasibility of the proposed algorithm.     

THE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON

In this paper,we address the problem of exact computation of the Hausdorff measure of a class of Sierpinski carpets E the self-similar sets generating in a unit regular pentagon on the plane.Under some conditions,we show the natural covering is the best one,and the Hausdorff measures of those sets are euqal to | E | s,where s=dim H E.     

A Level Set Representation Method for N-Dimensional Convex Shape and Applications

In this work,we present a new method for convex shape representation,which is regardless of the dimension of the concerned objects,using level-set approaches.To the best of our knowledge,the proposed prior is the first one which can work for high dimensional objects.Convexity prior is very useful for object completion in computer vision.It is a very challenging task to represent high dimensional convex objects.In this paper,we first prove that the convexity of the considered object is equivalent to the convexity of the associated signed distance function.Then,the second order condition of convex functions is used to characterize the shape convexity equivalently.We apply this new method to two applications:object segmentation with convexity prior and convex hull problem(especially with outliers).For both applications,the involved problems can be written as a general optimization problem with three constraints.An algorithm based on the alternating direction method of multipliers is presented for the optimization problem.Numerical experiments are conducted to verify the effectiveness of the proposed representation method and algorithm.     

Inequalities on the Triangle

The article deals with generalizations of the inequalities for convex functions on the triangle. The Jensen and the Hermite-Hadamard inequality are included in the study. Considering a convex function on the triangle, we obtain a generalization of the Jensen-Mercer inequality, and a refinement of the Hermite-Hadamard inequality.     

Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications

This paper is concerned with the study of optimality conditions for minimax optimization problems with an infinite number of constraints,denoted by(MMOP).More precisely,we first establish necessary conditions for optimal solutions to the problem(MMOP)by means of employing some advanced tools of variational analysis and generalized differentiation.Then,sufficient conditions for the existence of such solutions to the problem(MMOP)are investigated with the help of generalized convexity functions defined in terms of the limiting subdifferential of locally Lipschitz functions.Finally,some of the obtained results are applied to formulating optimality conditions for weakly efficient solutions to a related multiobjective optimization problem with an infinite number of constraints,and a necessary optimality condition for a quasiε-solution to problem(MMOP).     

An entropy based central cutting plane algorithm for convex min-max semi-infinite programming problems

In this paper,we present a central cutting plane algorithm for solving convex min-max semi-infinite programming problems.Because the objective function here is non-differentiable,we apply a smoothing technique to the considered problem and develop an algorithm based on the entropy function.It is shown that the global convergence of the proposed algorithm can be obtained under weaker conditions.Some numerical results are presented to show the potential of the proposed algorithm.     

Time variant multi-objective linear fractional interval-valued transportation problem

This paper studies a time-variant multi-objective linear fractional transportation problem. In reality, transported goods should reach in destinations within a specific time. Considering the importance of time, a time-variant multi-objective linear fractional transportation problem is formulated here. We take into account the parameters as cost, supply and demand are interval valued that involved in the proposed model, so we treat the model as a multi-objective linear fractional interval transportation problem. To solve the formulated model, we first convert it into a deterministic form using a new transformation technique and then apply fuzzy programming to solve it. The applicability of our proposed method is shown by considering two numerical examples. At last, conclusions and future research directions regarding our study is included.     

HGAH矩阵的逆特征值问题

In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamihonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.     

基于DC规划方法的稀疏临近支持向量机

为了提高临近支持向量机（PSVM）的数值表现,在PSVM的模型中引入了$\ell_0$-范数正则项,提出了稀疏临近支持向量机模型（SPSVM）,从而提高分类器的特征选择能力。然而带有$\ell_0$-范数正则项的问题往往是NP-难问题,为了克服这一问题,采用非凸连续函数近似$\ell_0$-范数,并通过适当的DC分解将问题转化成DC规划问题进行求解,同时还讨论了算法的收敛性。数值实验结果表明不论是在仿真数据还是在实际数据中,所提出的方法是比较有效稳定的。     

Higher Order Optimality Conditions in Nonsmooth Optimization

For an arbitrary function $\font\Opr=msbm10 at 8pt \def\Op#1{\hbox {\Opr{#1}}}f\!\!: {\bf E} \rightarrow {\bar {\Op R}}$     

Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation

Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixed-integer programming for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [21] and Ceria and Soares [6], we propose a convex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a Mixed-Integer Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional big-M formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.     

Remarks on Risk-Sensitive Control Problems

The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor $\alpha$ goes to zero. If $u_\alpha(\theta,x)$ denotes the optimal cost function, $\theta$ being the risk factor, then it is shown that $\lim_{\alpha\to 0}\alpha u_\alpha(\theta,x)=\xi(\theta)$ where $\xi(\theta)$ is the average on $]0,\theta[$ of the optimal cost of the (usual) infinite horizon risk-sensitive control problem.     

Methods of Chebyshev points of convex sets and their applications

Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.     

交替最小化算法求解强凸函数与弱凸函数和的极小值问题

交替最小化算法(简称AMA)最早由[SIAM J.Control Optim.,1991,29(1):119-138]提出,并能用于求解强凸函数与凸函数和的极小值问题.本文直接利用AMA算法来求解强凸函数与弱凸函数和的极小值问题.在强凸函数的模大于弱凸函数的模的假设下,我们证明了AMA生成的点列全局收敛到优化问题的解,并且若该优化问题中的某个函数是光滑函数时,AMA所生成的点列的收敛率是线性的.     

一类特殊DC规划的对偶性

根据共轭函数和DC规划的性质，给出一类特殊DC规划的共轭对偶并讨论其对偶规划的特殊性质，然后利用该性质，把对这类特殊DC规划的求解转化为对一个凸规划的求解。     

Composite geometric programming

Geometric programming is based on functions called posynomials, the terms of which are log-linear. This class of programs is extended from the composition of an exponential and a linear function to an exponential and a convex function. The resulting duality theory for composite geometric programs retains many of the qualities of geometric programming duality, while at the same time encompassing new areas of application. As an application, composite geometric programming is applied to exponential geometric programming. A pure dual is developed for the first time and used to solve a problem from the literature.This research was supported by the Air Force Office of Scientific Research, Grant No. AFOSR-83-0234.     

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

We describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation.     

Deriving an unconstrained convex program for linear programming

Consider a linear programming problem in Karmarkar's standard form. By perturbing its linear objective function with an entropic barrier function and applying generalized geometric programming theory to it, Fang recently proposed an unconstrained convex programming approach to finding an epsilon-optimal solution. In this paper, we show that Fang's derivation of an unconstrained convex dual program can be greatly simplified by using only one simple geometric inequality. In addition, a system of nonlinear equations, which leads to a pair of primal and dual epsilon-optimal solutions, is proposed for further investigation.This work was partially supported by the North Carolina Supercomputing Center and a 1990 Cray Research Grant. The authors are indebted to Professors E. L. Peterson and R. Saigal for stimulating discussions.