This paper treats entropy constrained linear programs from modelling as well as computational aspects. The optimal solutions to linear programs with one additional entropy constraint are expressed in terms of Lagrange...
详细信息
This paper treats entropy constrained linear programs from modelling as well as computational aspects. The optimal solutions to linear programs with one additional entropy constraint are expressed in terms of Lagrange-multipliers. Conditions for uniqueness are given. Sensitivity and duality are studied. The Newton—Kantorovich method is used to obtain a locally convergent iterative procedure. Related problems based on maximum entropy or minimum information are discussed.
This paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function. The paper contains extensions and ana...
详细信息
This paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function. The paper contains extensions and analysis of two polynomial-time algorithms for linear programming. We first present an extension of Gonzaga's O(nL) iteration algorithm, that computes dual variables and does not assume a known optimal objective function value. This algorithm uses only affine scaling, and is based on computing the projected gradient of the potential function [GRAPHICS] where x is the vector of primal variables and s is the vector of dual slack variables, and q = n + square-root n. The algorithm takes either a primal step or recomputes dual variables at each iteration. We next present an alternate form of Ye's O(square-root n L) iteration algorithm, that is an extension of the first algorithm of the paper, but uses the potential function [GRAPHICS] where q = n + square-root n. We use this alternate form of Ye's algorithm to show that Ye's algorithm is optimal with respect to the choice of the parameter q in the following sense. Suppose that q = n + n(t) where t greater-than-or-equal-to 0. Then the algorithm will solve the linear program in O(n(r)L) iterations, where r = max{t, 1-t}. Thus the value of t that minimizes the complexity bound is t = 1/2, yielding Ye's O(square-root n L) iteration bound.
We propose a new approach to combine linear programming (LP) interior-point and simplex pivoting algorithms. In any iteration of an interior-point algorithm we construct a related LP problem, which approximates the or...
详细信息
We propose a new approach to combine linear programming (LP) interior-point and simplex pivoting algorithms. In any iteration of an interior-point algorithm we construct a related LP problem, which approximates the original problem, with a known (strictly) complementary primal-dual solution pair. Thus, we can apply Megiddo's (1991) pivoting procedure to compute an optimal basis for the approximate problem in strongly polynomial time. We show that, if the approximate problem is constructed from an interior-point iterate sufficiently close to the optimal face, then any optimal basis of the approximate problem is an optimal basis for the original problem. If the LP data are rational, the total number of interior-point iterations to create such a sufficient approximate problem is bounded by a polynomial in the data size. We develop a modification of Megiddo's procedure and discuss several implementation issues in solving the approximate problem. We also report encouraging computational results for this combined approach.
The focal point of this paper is the probabilistically constrained linear program ( PCLP) and how it can be applied to control system design under risk constraints. The PCLP is the counterpart of the classical linear ...
详细信息
The focal point of this paper is the probabilistically constrained linear program ( PCLP) and how it can be applied to control system design under risk constraints. The PCLP is the counterpart of the classical linear program, where it is assumed that there is random uncertainty in the constraints and, therefore, the deterministic constraints are replaced by probabilistic ones. It is shown that for a wide class of probability density functions, called log- concave symmetric densities, the PCLP is a convex program. An equivalent formulation of the PCLP is also presented which provides insight into numerical implementation. This concept is applied to control system design. It is shown how the results in this paper can be applied to the design of controllers for discrete- time systems to obtain a closed loop system with a well- defined risk of violating the so- called property of superstability. Furthermore, we address the problem of risk- adjusted pole placement.
Mangasarian (Optim. Lett., 6(3), 431-436, 2012) proposed a constraints transformation based approach to securely solving the horizontally partitioned linear programs among multiple entities-every entity holds its own ...
详细信息
Mangasarian (Optim. Lett., 6(3), 431-436, 2012) proposed a constraints transformation based approach to securely solving the horizontally partitioned linear programs among multiple entities-every entity holds its own private equality constraints. More recently, Li et al. (Optim. Lett., doi: 10.1007/s11590-011-0403-2, 2012) extended the transformation approach to horizontally partitioned linear programs with inequality constraints. However, such transformation approach is not sufficiently secure - occasionally, the privately owned constraints are still under high risk of inference. In this paper, we present an inference-proof algorithm to enhance the security for privacy-preserving horizontally partitioned linear program with arbitrary number of equality and inequality constraints. Our approach reveals significantly less information than the prior work and resolves the potential inference attack.
We present a smoothing-type algorithm for solving the linear program (LP) by making use of an augmented system of its optimality conditions. The algorithm is shown to be globally convergent without requiring any assum...
详细信息
We present a smoothing-type algorithm for solving the linear program (LP) by making use of an augmented system of its optimality conditions. The algorithm is shown to be globally convergent without requiring any assumption. It only needs to solve one system of linear equations and to perform one line search at each iteration. In particular, if the LP has a solution (and hence it has a strictly complementary solution), then the algorithm will generate a strictly complementary solution of the LP;and if the LP is infeasible, then the algorithm will correctly detect infeasibility of the LP. To the best of our knowledge, this is the first smoothing-type algorithm for solving the LP having the above desired convergence features. (c) 2005 Elsevier Inc. All rights reserved.
We present a convergence analysis for a class of inexact infeasible-interior-point methods for solving linear programs. The main feature of inexact methods is that the linear systems defining the search direction at e...
详细信息
We present a convergence analysis for a class of inexact infeasible-interior-point methods for solving linear programs. The main feature of inexact methods is that the linear systems defining the search direction at each interior-point interation need not be solved to high accuracy More precisely, we allow that these linear systems are only solved to a moderate accuracy in the residual, but no assumptions are made on the accuracy of the search direction in the search space. In particular, our analysis does not require that feasibility is maintained even if the initial iterate happened to be a feasible solution of the linear program. We also present a few numerical examples to illustrate the effect of using inexact search direction on the number of interior-point iterations.
A parallel method for globally minimizing a linear program with an additional reverse convex constraint is proposed which combines the outer approximation technique and the cutting plane method. Basically p (less than...
详细信息
A parallel method for globally minimizing a linear program with an additional reverse convex constraint is proposed which combines the outer approximation technique and the cutting plane method. Basically p (less than or equal to n) processors are used for a problem with a variables and a globally optimal solution is found effectively in a finite number of steps. Computational results are presented for test problems with a number of variables up to 80 and 63 linear constraints (plus nonnegativity constraints). These results were obtained on a distributed-memory MIMD parallel computer, DELTA, by running both serial and parallel algorithms with double precision. Also, based on 40 randomly generated problems of the same size, with 16 variables and 32 linear constraints (plus x greater than or equal to 0), the numerical results from different number processors are reported, including the serial algorithm's. (C) 1997 Academic Press.
We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief ...
详细信息
We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. We also show that when the LP relaxation has a fractional solution then the BP algorithm can be used to solve the LP relaxation. Our proof is based on the notion of graph covers and extends the analyses of [M. Bayati, D. Shah, and M. Sharma, in Proceedings of the IEEE Int. Symp. Information Theory, 2005] and [B. Huang and T. Jebara, in Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, 2007]. The result is notable in the following regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure;(2) Variants of the proof work for both synchronous and asynchronous BP;it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem.
This article shows how to solve linear programs of the form min(Ax=b,x) (>= 0) c(inverted perpendicular)x with n variables in time O*((n(omega) + n(2.5-alpha/2) + n(2+1/6)) log(n/delta)), where omega is the exponen...
详细信息
This article shows how to solve linear programs of the form min(Ax=b,x) (>= 0) c(inverted perpendicular)x with n variables in time O*((n(omega) + n(2.5-alpha/2) + n(2+1/6)) log(n/delta)), where omega is the exponent of matrix multiplication, alpha is the dual exponent of matrix multiplication, and delta is the relative accuracy. For the current value of omega similar to 2.37 and alpha similar to 0.31, our algorithm takes O* (n(omega)log(n/delta)) time. When omega = 2, our algorithm takes O* (n(2+1/6) log(n/delta)) time. Our algorithm utilizes several new concepts that we believe may be of independent interest: We define a stochastic central path method. We show how to maintain a projection matrix root WA(inverted perpendicular) (AWA(inverted perpendicular))(-1)A root W in sub-quadratic time under l(2) multiplicative changes in the diagonal matrix W.
暂无评论