Markov decision processes for infinite horizon problems solved with the cosine simplex method

This article presents a new method of linear programming (LP) for solving Markov decision processes (MDPs) based on the simplex method (SM). SM has shown to be the most efficient method in many practical problems; unfortunately, classical SM has an exponential complexity. Therefore, new SMs have emerged for obtaining optimal solutions in the most efficient way. The cosine simplex method (CSM) is one of them. CSM is based on the Karush Kuhn Tucker conditions, and is able to efficiently solve general LP problems. This work presents a new method named the Markov Cosine Simplex Method (MCSM) for solving MDP problems, which is an extension of CSM. In this article, the efficiency of MCSM is compared to the traditional revised simplex method (RSM); experimental results show that MCSM is far more efficient than RSM.     

确定线性规划全部最优解的方法

使用凸多面体的表示定理 ,导出了标准型线性规划最优解的一般表达式 ,并基于单纯形法 ,给出最优解唯一性条件以及当唯一性条件不满足时求出全部最优解的计算步骤 ,同时附有数值例子 .     

The double pivot simplex method

The simplex method, created by George Dantzig, optimally solves a linear program by pivoting. Dantzig’s pivots move from a basic feasible solution to a different basic feasible solution by exchanging exactly one basic variable with a nonbasic variable. This paper introduces the double pivot simplex method, which can transition between basic feasible solutions using two variables instead of one. Double pivots are performed by identifying the optimal basis in a two variable linear program using a new method called the slope algorithm. The slope algorithm is fast and allows an iteration of DPSM to have the same theoretical running time as an iteration of the simplex method. Computational experiments demonstrate that DPSM decreases the average number of pivots by approximately 41% on a small set of benchmark instances.     

Efficient enumeration of the vertices of polyhedra associated with network LP's

Algorithms are given to list the vertices of polyhedra associated with network linear programs and their duals. Each algorithm has running time which is quadratic in the number of vertices of the polyhedron, and does not require that the polyhedron be either bounded or simple. The algorithms use characterizations of adjacent vertices in network and dual network LP's to perform an efficient traversal of the edge graph of the polyhedron. This contrasts with algorithms for enumerating the vertices of a general polyhedron, all of which have worst-case complexity which is exponential in the number of vertices of the polyhedron.     

Parallel search paths for the simplex algorithm

It is well known that the simplex method is inherently a sequential algorithm with little scope for parallelization. Even so, during the last decades several attempts were made to parallelize it since it is one of the most important algorithms for solving linear optimization problems. Such parallelization ideas mostly rely on iteration parallelism and overlapping. Since the simplex method goes through a series of basic solutions until it finds an optimal solution, each of them must be available before performing the next basis change. This phenomenon imposes a limit on the performance of the parallelized version of the simplex method which uses overlapping iterations. Another approach can be considered if we think about alternative paths on the n-dimensional simplex polyhedron. As the simplex method goes through the edges of this polyhedron it is generally true that the speed of convergence of the algorithm is not smooth. It depends on the actual part of the surface. If a parallel version of the simplex algorithm simultaneously goes on different paths on this surface a highly reliable algorithm can be constructed. There is no known dominating strategy for pivot selection. Therefore, one can try different pivot selection methods in parallel in order to guide the algorithm on different pathways. This approach can be used effectively with periodic synchronization on shared memory multi-core computing environments to speed up the solution algorithm and get around numerically and/or algorithmically difficult situations throughout the computations.     

A new LQP alternating direction method for solving variational inequality problems with separable structure

We presented a new logarithmic-quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The predictor is obtained by solving series of related systems of non-linear equations in a parallel wise. The new iterate is obtained by searching the optimal step size along a new descent direction. The new direction is obtained by the linear combination of two descent directions. Global convergence of the proposed method is proved under certain assumptions. We show the O(1 / t) convergence rate for the parallel LQP alternating direction method.     

ON AN EFFICIENT IMPLEMENTATION OF THE FACE ALGORITHM FOR LINEAR PROGRAMMING＊

In this paper, we consider the solution of the standard linear programming [Lt＇）. A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The face algorithm proposed by Pan [19] targets at the optimal face by iterating from face to face, along an orthogonal projection of the negative objective gradient onto a relevant null space. The algorithm exhibits a favorable numerical performance by comparing the simplex method. In this paper, we further investigate the face algorithm by proposing an improved implementation. In exact arithmetic computation, the new algorithm generates the same sequence as Pan＇s face algorithm, but uses less computational costs per iteration, and enjoys favorable properties for sparse problems.     

On the Equivalence of the Simplex Methods and a Multiplier-Alike Method for Linear Programming

In linear programming, the simplex method has been viewed for a long time as an efficient tool. Interior methods have attracted a lot of attention since they were proposed recently. It seems plausible intuitively that there is no reason why a good linear programming algorithm should not be allowed to cross the boundary of the feasible region when necessary. However, such an algorithm is seldom studied. In this paper, we will develop first a framework of a multiplier-alike algorithm for linear programming which allows its trajectory to move across the boundary of the feasible region. Second, we illustrate that such a framework has the potential to perform as well as the simplex method by showing that these methods are equivalent in a well-defined sense, even though they look so different.     

An implementation of steepest-descent augmentation for linear programs

Generalizing the simplex method, circuit augmentation schemes for linear programs follow circuit directions through the interior of the underlying polyhedron. Steepest-descent augmentation is especially promising, but an implementation of the iterative scheme is a significant challenge. We work towards a viable implementation through a model in which a single linear program is updated dynamically to remain in memory throughout. Computational experiments exhibit dramatic improvements over a naïve approach and reveal insight into the next steps required for large-scale computations.     

Enumerative technique for an extreme point fractional program

This paper presents a technique for solving a linear fractional functionals program whose optimum is supposed to occur at one of the extreme points of a convex polyhedron. This polyhedron is defined by a system of linear inequalities with non-negative restrictions on the variables. The approach is a dual method type in the sense that feasibility is achieved only at the optimal solution.     

二阶锥规划的基于自协调指数核函数的原始-对偶内点算法

基于一个自协调指数核函数, 设计求解二阶锥规划的原始-对偶内点算法. 根据自协调指数核函数的二阶导数与三阶导数的特殊关系, 在求解问题的中心路径时, 用牛顿方向代替了负梯度方向来确定搜索方向. 由于自协调指数核函数不具有``Eligible'性质, 在分析算法的迭代界时, 利用牛顿方法求解目标函数满足自协调性质的无约束优化问题的技术, 估计算法内迭代中自协调指数核函数确定的障碍函数的下降量, 得到原始-对偶内点算法大步校正的迭代界O(2N\frac{\log2N}{\varepsilon}), 这里N是二阶锥的个数. 这个迭代界与线性规划情形下的迭代界一致. 最后, 通过数值算例验证了算法的有效性.     

A first-order interior-point method for linearly constrained smooth optimization

We propose a first-order interior-point method for linearly constrained smooth optimization that unifies and extends first-order affine-scaling method and replicator dynamics method for standard quadratic programming. Global convergence and, in the case of quadratic program, (sub)linear convergence rate and iterate convergence results are derived. Numerical experience on simplex constrained problems with 1000 variables is reported.     

Worst case behavior of the steepest edge simplex method

At each nondegenerate iteration of the steepest-edge simplex method one moves from a vertex of the polyhedron, P, of feasible points to an adjacent vertex along an edge that is steepest with respect to the linear objective function ψ. In this paper we show how to construct a sequence of linear programs (P_n,ψ_n) in n variables for which the number of iterations required by the steepest edge simplex method is 2ⁿ−1.     

Reminiscences about the origins of linear programming

The author recalls the early days of linear programming, the contributions of von Neumann, Leontief, Koopmans and others.Linear Programming is viewed as a revolutionary development giving us the ability for the first time to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity.     

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC^1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subproblem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.     

变量有广义界线性规划的直接对偶单纯形法

本文讨论变量有广义界线性规划问题借助标准形线性规划同单纯形法技术，建立问题的一个直接对偶单纯形法。分析了方法的性质，给出了初始对偶可行基的计算方法，并用实例说明方法的具体操作。     

A feasible direction method for the semidefinite program with box constraints

In this paper, we try to solve the semidefinite program with box constraints. Since the traditional projection method for constrained optimization with box constraints is not suitable to the semidefinite constraints, we present a new algorithm based on the feasible direction method. In the paper, we discuss two cases: the objective function in semidefinite programming is linear and nonlinear, respectively. We establish the convergence of our algorithm, and report the numerical experiments which show the effectiveness of the algorithm.