A parameterized proximal point algorithm for separable convex optimization

In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1 / t) convergence rate, where t denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of multipliers and the relaxed proximal point algorithm.     

一类线性乘积规划问题的分支定界缩减方法

提出了求解一类线性乘积规划问题的分支定界缩减方法, 并证明了算法的收敛性.在这个方法中, 利用两个变量乘积的凸包络技术, 给出了目标函数与约束函数中乘积的下界, 由此确定原问题的一个松弛凸规划, 从而找到原问题全局最优值的下界和可行解. 为了加快所提算法的收敛速度, 使用了超矩形的缩减策略. 数值结果表明所提出的算法是可行的.     

A finite steps algorithm for solving convex feasibility problems

This paper develops a new variant of the classical alternating projection method for solving convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space. An extension to the feasibility problem for the intersection of two convex sets is presented as well. It is shown that one can solve such problems in a finite number of steps and an explicit upper bound for the required number of steps is obtained. As an application, we propose a new finite steps algorithm for linear programming with linear matrix inequality constraints. This solution is computed by solving a sequence of a matrix eigenvalue decompositions. Moreover, the proposed procedure takes advantage of the structure of the problem. In particular, it is well adapted for problems with several small size constraints.     

ε-Optimal solutions in nondifferentiable convex programming and some related questions

In this paper we present ε-optimality conditions of the Kuhn-Tucker type for points which are within ε of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints, linear equality constraints and abstract constraints. Such ε-optimality conditions are of interest for theoretical consideration as well as from the computational point of view. Some illustrative applications are made. Thus we derive an expression for the ε-subdifferential of a general convex ‘max function’. We also show how the ε-optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints.     

A trust region algorithm for solving bilevel programming problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.     

The Complex Interior-Boundary method for linear and nonlinear programming with linear constraints

In this paper we develop the Complex method; an algorithm for solving linear programming (LP) problems with interior search directions. The Complex Interior-Boundary method (as the name suggests) moves in the interior of the feasible region from one boundary point to another of the feasible region bypassing several extreme points at a time. These directions of movement are guaranteed to improve the objective function. As a result, the Complex method aims to reach the optimal point faster than the Simplex method on large LP programs. The method also extends to nonlinear programming (NLP) with linear constraints as compared to the generalized-reduced gradient.The Complex method is based on a pivoting operation which is computationally efficient operation compared to some interior-point methods. In addition, our algorithm offers more flexibility in choosing the search direction than other pivoting methods (such as reduced gradient methods). The interior direction of movement aims at reducing the number of iterations and running time to obtain the optimal solution of the LP problem compared to the Simplex method. Furthermore, this method is advantageous to Simplex and other convex programs in regard to starting at a Basic Feasible Solution (BFS); i.e. the method has the ability to start at any given feasible solution.Preliminary testing shows that the reduction in the computational effort is promising compared to the Simplex method.     

A method of linearizations for linearly constrained nonconvex nonsmooth minimization

A readily implementable algorithm is given for minimizing a (possibly nondifferentiable and nonconvex) locally Lipschitz continuous functionf subject to linear constraints. At each iteration a polyhedral approximation tof is constructed from a few previously computed subgradients and an aggregate subgradient, which accumulates the past subgradient information. This aproximation and the linear constraints generate constraints in the search direction finding subproblem that is a quadratic programming problem. Then a stepsize is found by an approximate line search. All the algorithm's accumulation points are stationary. Moreover, the algorithm converges whenf happens to be convex.     

Linear multiplicative programming

An algorithm for solving a linear multiplicative programming problem (referred to as LMP) is proposed. LMP minimizes the product of two linear functions subject to general linear constraints. The product of two linear functions is a typical non-convex function, so that it can have multiple local minima. It is shown, however, that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure. The computational results indicate that the amount of computation is not much different from that of solving linear programs of the same size. In addition, the method proposed for LMP can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.     

Globally and superlinearly convergent trust-region algorithm for convex SC1-minimization problems and its application to stochastic programs

A function mapping from ⁿ to is called an SC¹-function if it is differentiable and its derivative is semismooth. A convex SC¹-minimization problem is a convex minimization problem with an SC¹-objective function and linear constraints. Applications of such minimization problems include stochastic quadratic programming and minimax problems. In this paper, we present a globally and superlinearly convergent trust-region algorithm for solving such a problem. Numerical examples are given on the application of this algorithm to stochastic quadratic programs.This work was supported by the Australian Research Council.We are indebted to Dr. Xiaojun Chen for help in the computation. We are grateful to two anonymous referees for their comments and suggestions, which improved the presentation of this paper.     

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.     

交替方向法求解带线性约束的变分不等式

１引言变分不等式是一个有广泛应用的数学问题,它的一般形式是：确定一个向量,使其满足这里ｆ是一个从到自身的一个映射,Ｓ是Ｒ中的一个闭凸集．在许多实际问题中集合Ｓ往往具有如下结构其中ＡｂＫ是中的一个简单闭凸集．例如一个正卦限,一个框形约束结构,或者一个球简言之,Ｓ是Ｒ中的一个超平面与一个简单闭凸集的交．求解问题（１）－（２）,往往是通过对线性约束Ａ引人Ｌａｇｒａｎｇｅ乘子,将原问题化为如下的变分不等式：确定使得我们记问题（３）－（４）为ＶＩ（Ｆ）．熟知［３］,ＶＩ（,Ｆ）等价于投影方程其中凡（·）表…     

Piecewise-linear programming: The compact (CPLP) algorithm

A compact algorithm is presented for solving the convex piecewise-linear-programming problem, formulated by means of a separable convex piecewise-linear objective function (to be minimized) and a set of linear constraints. This algorithm consists of a finite sequence of cycles, derived from the simplex method, characteritic of linear programming, and the line search, characteristic of nonlinear programming. Both the required storage and amount of calculation are reduced with respect to the usual approach, based on a linear-programming formulation with an expanded tableau. The tableau dimensions arem×(n+1), wherem is the number of constraints andn the number of the (original) structural variables, and they do not increase with the number of breakpoints of the piecewise-linear terms constituting the objective function.     

Analytic efficient solution set for multi-criteria quadratic programs

We present active set methods to evaluate the exact analytic efficient solution set for multi-criteria convex quadratic programming problems (MCQP) subject to linear constraints. The idea is based on the observations that a strictly convex programming problem admits a unique solution, and that the efficient solution set for a multi-criteria strictly convex quadratic programming problem with linear equality constraints can be parameterized. The case of bi-criteria quadratic programming (BCQP) is first discussed since many of the underlying ideas can be explained much more clearly in the case of two objectives. In particular we note that the efficient solution set of a BCQP problem is a curve on the surface of a polytope. The extension to problems with more than two objectives is straightforward albeit some slightly more complicated notation. Two numerical examples are given to illustrate the proposed methods.     

求线性比式和问题全局解的输出空间分枝定界算法

基于对p-1维输出空间进行剖分的思想,提出了一种求解线性比式和问题的分枝定界算法.通过一种两阶段转换方法得到原问题的一个等价问题,该问题的非凸性主要体现在新增加的p-1个非线性等式约束上.利用双线性函数的凹凸包络对这些非线性约束进行凸化,这就为等价问题构造了凸松弛子问题.将凸松弛子问题中的冗余约束去掉并进行等价转换,从而获得了一个比凸松弛子问题规模更小、约束更少的线性规划问题.证明了算法的理论收敛性和计算复杂性.数值实验表明该算法是有效可行的.     

Convex programming with single separable constraint and bounded variables

In this paper a minimization problem with convex objective function subject to a separable convex inequality constraint “≤” and bounded variables (box constraints) is considered. We propose an iterative algorithm for solving this problem based on line search and convergence of this algorithm is proved. At each iteration, a separable convex programming problem with the same constraint set is solved using Karush-Kuhn-Tucker conditions. Convex minimization problems subject to linear equality/ linear inequality “≥” constraint and bounds on the variables are also considered. Numerical illustration is included in support of theory.     

Fast value iteration: an application of Legendre-Fenchel duality to a class of deterministic dynamic programming problems in discrete time*

ABSTRACT

We propose an algorithm, which we call ‘Fast Value Iteration’ (FVI), to compute the value function of a deterministic infinite-horizon dynamic programming problem in discrete time. FVI is an efficient algorithm applicable to a class of multidimensional dynamic programming problems with concave return (or convex cost) functions and linear constraints. In this algorithm, a sequence of functions is generated starting from the zero function by repeatedly applying a simple algebraic rule involving the Legendre-Fenchel transform of the return function. The resulting sequence is guaranteed to converge, and the Legendre-Fenchel transform of the limiting function coincides with the value function.     

求解具有确定性线性不等式约束的有补偿随机规划问题的近似ε——容许方向法

1引言形如线性不等式确定性约束的有补偿随机规划问题     

On an algorithm solving two-level programming problems with nonunique lower level solutions

In the paper, an algorithm is presented for solving two-level programming problems. This algorithm combines a direction finding problem with a regularization of the lower level problem. The upper level objective function is included in the regularzation to yield uniqueness of the follower's solution set. This is possible if the problem functions are convex and the upper level objective function has a positive definite Hessian. The computation of a direction of descent and of the step size is discussed in more detail. Afterwards the convergence proof is given.Last but not least some remarks and examples describing the difficulty of the inclusion of upper-level constraints also depending on the variables of the lower level are added.     

Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs

In this article, a branch and-bound outer approximation algorithm is presented for globally solving a sum-of-ratios fractional programming problem. To solve this problem, the algorithm instead solves an equivalent problem that involves minimizing an indefinite quadratic function over a nonempty, compact convex set. This problem is globally solved by a branch-and-bound outer approximation approach that can create several closed-form linear inequality cuts per iteration. In contrast to pure outer approximation techniques, the algorithm does not require computing the new vertices that are created as these cuts are added. Computationally, the main work of the algorithm involves solving a sequence of convex programming problems whose feasible regions are identical to one another except for certain linear constraints. As a result, to solve these problems, an optimal solution to one problem can potentially be used to good effect as a starting solution for the next problem.     

A parametric successive underestimation method for convex multiplicative programming problems

This paper addresses itself to the algorithm for minimizing the product of two nonnegative convex functions over a convex set. It is shown that the global minimum of this nonconvex problem can be obtained by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in a higher dimensional space and to apply a branch-and-bound algorithm using an underestimating function. Computational results indicate that our algorithm is efficient when the objective function is the product of a linear and a quadratic functions and the constraints are linear. An extension of our algorithm for minimizing the sum of a convex function and a product of two convex functions is also discussed.