Extending our previous work [T. Wang, R. D. C. Monteiro, and J.-S. Pang, Math. Programming, 74 (1996), pp. 159-195], this paper presents a general potentialreduction Newton method for solving a constrained system of ...
详细信息
Extending our previous work [T. Wang, R. D. C. Monteiro, and J.-S. Pang, Math. Programming, 74 (1996), pp. 159-195], this paper presents a general potentialreduction Newton method for solving a constrained system of nonlinear equations. A major convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed. Strengthened convergence results are established in the context of these specializations.
In this work, we study several extensions of the potential reduction algorithm that was developed for linear programming. These extensions include choosing different potential functions, generating the analytic center...
详细信息
In this work, we study several extensions of the potential reduction algorithm that was developed for linear programming. These extensions include choosing different potential functions, generating the analytic center of a polytope, and finding the equilibrium of a zero-sum bimatrix game.
In this short note, we prove that the global convergence rate of the duality gap in a symmetric primal-dual potentialalgorithm for linear programming, without line search, is no better than linear. More specifically,...
详细信息
In this short note, we prove that the global convergence rate of the duality gap in a symmetric primal-dual potentialalgorithm for linear programming, without line search, is no better than linear. More specifically, the convergence rate is no better than (1 - tau/square-root n), where tau is a number between zero and one.
This paper presents a wide class of globally convergent interior-point algorithms for the nonlinear complementarity problem with a continuously differentiable monotone mapping in terms of a unified global convergence ...
详细信息
This paper presents a wide class of globally convergent interior-point algorithms for the nonlinear complementarity problem with a continuously differentiable monotone mapping in terms of a unified global convergence theory given by Polak in 1971 for general nonlinear programs. The class of algorithms is characterized as: Move in a Newton direction for approximating a point on the path of centers of the complementarity problem at each iteration. Starting from a strictly positive but infeasible initial point, each algorithm in the class either generates an approximate solution with a given accuracy or provides us with information that the complementarity problem has no solution in a given bounded set. We present three typical examples of our interior-point algorithms, a horn neighborhood model, a constrained potentialreduction model with the use of the standard potential function, and a pure potentialreduction model with the use of a new potential function.
This paper proposes an interior point algorithm for a positive semi-definite linear complementarity problem: find an (x,y) is-an-element-of-R2n such that y = Mx + q, (x,y) greater-than-or-equal-to 0 and x(T)y = 0. The...
详细信息
This paper proposes an interior point algorithm for a positive semi-definite linear complementarity problem: find an (x,y) is-an-element-of-R2n such that y = Mx + q, (x,y) greater-than-or-equal-to 0 and x(T)y = 0. The algorithm reduces the potential function [GRAPHICS] by at least 0.2 in each iteration requiring O(n3) arithmetic operations. If it starts from an interior feasible solution with the potential function bounded by O(square-root n L), it generates, in at most O(square-root n L) iterations, an approximate solution with the potential function value -O(square-root n L), from which we can compute an exact solution in O(n3) arithmetic operations. The algorithm is closely related with the central path following algorithm recently given by the authors. We also suggest a unified model for both potentialreduction and path following algorithms for positive semi-definite linear complementarity problems.
potential function reductionalgorithms for linear programming and the linear complementarity problem use key projections p(x) and p(s) which are derived from the 'double' potential function, phi(x, s) = rho l...
详细信息
potential function reductionalgorithms for linear programming and the linear complementarity problem use key projections p(x) and p(s) which are derived from the 'double' potential function, phi(x, s) = rho ln(x(T)s)-SIGMA(j = 1)n ln(x(j)s(j)), where x and s are primal and dual slacks vectors. For non-symmetric LP duality we show that the existence of sBAR, yBAR, xBAR satisfying sBAR = c - A(T)yBAR, AxBAR = b such that p(x) = (rho/x(T)s)XsBAR - e and p(s) = (rho/x(T)s)SxBAR - e yields simultaneous primal and dual projection-based updating during the process of reducing the potential function phi. The role of xBAR, sBAR in an O(square-root n L) simultaneous primal-dual update algorithm is discussed.
When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an art...
详细信息
When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an artificial problem having a known interior feasible solution from which we start the algorithm. The artificial problem involves a constant M (or constants) which we need to choose large enough to ensure the equivalence between the artificial problem and the original problem. Theoretically, we can always assign a positive number of the order O(2L) to M in linear cases, where L denotes the input size of the problem. Practically, however, such a large number is impossible to implement on computers. If we choose too large M, we may have numerical instability and/or computational inefficiency, while the artificial problem with M not large enough will never lead to any solution of the original problem. To solve this difficulty, this paper presents ''a little theorem of the big M'', which will enable us to find whether M is not large enough, and to update M during the iterations of the algorithm even if we start with a smaller M. Applications of the theorem are given to a polynomial-time potential reduction algorithm for positive semi-definite linear complementarity problems, and to an artificial self-dual linear program which has a close relation with the primal-dual interior point algorithm using Lustig's limiting feasible direction vector.
A potential reduction algorithm is developed for solving entropy optimization problems. It is shown that the algorithm generates an epsilon-optimal solution within at most O(root n vertical bar log epsilon vertical ba...
详细信息
A potential reduction algorithm is developed for solving entropy optimization problems. It is shown that the algorithm generates an epsilon-optimal solution within at most O(root n vertical bar log epsilon vertical bar) iterations, where, as usual, n is the number of nonnegative variables, and each iteration solves a system of linear equations. Under a computable criterion, the algorithm is tuned to the pure Newton method in a manner that leads to quadratic convergence while maintaining primal feasibility at each step. A stopping criterion is derived which ensures that the objective function approaches its optimal value within any prescribed tolerance. This applies for all entropy optimization problems having interior optimal solutions.
This paper describes an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the "projectio...
详细信息
This paper describes an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the "projection" at each iteration is performed by using a primal-dual interior-point method for convex quadratic programming. Convergence properties can be maintained even if the projection is done inexactly in a well-defined way. Higher-order derivative information on the manifold defined by the apparently active constraints can be used to increase the rate of local convergence.
We analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions. We show that, under a strong probabilistic assumption regarding the distribution of the data in an iter...
详细信息
We analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions. We show that, under a strong probabilistic assumption regarding the distribution of the data in an iteration, the decrease in the primal potential function will be OMEGA(rho/square-root log(n)) with high probability, compared to the guaranteed OMEGA(1). (rho greater-than-or-equal-to 2n is a parameter in the potential function and n is the number of variables.) Under the same assumption, we further show that the objective reduction rate of Dikin's affine scaling algorithm is (1 - 1/square-root log(n)) with high probability, compared to no guaranteed convergence rate.
暂无评论