Self-dual Problem Research Articles

Symmetric duality with pseudo-invexity in variational problems

Weak and strong duality results are established under pseudo-invexity hypotheses for symmetric dual variational problems. Self-dual problems and static symmetric dual programs are included as special cases.

Conic convex programming and self-dual embedding

How ro initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the two-phase approach or using the so-called big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem. Viz. the self-dual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving tts embedded self-dual problem The embedded self-dual problem has a trivial initial solution and has the same stmctare as the original problem. Hence. it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programmng. Since a nonlinear conic convex programming problem may lack the so-called strict complementarity property, it canses diffcairier in identifying solutions for the original problem; based on solutions for the embedded self-dual system. We provide numerous examples from semidefinite programming to illustrate various possibilities which have no analogue in the linear programming case.

On the topological theory of self-dual connections over noncompact Riemann surfaces

A topological action for self-dual connections over noncompact Riemann surfaces is proposed. TheJ formulation and the associated linear system are obtained. A new connection is constructed, depending on a Kac-Moody parameter such that its flatness condition is theJ-equation associated to the self-dual problem. The algebra of infinitesimal Backlund transformations depending on this Kac-Moody parameter is constructed.

A Surface of Analytic Centers and Primal-Dual Infeasible-Interior-Point Algorithms for Linear Programming

We define a surface of analytic centers determined by a primal-dual pair of linear programming problems and an infeasible interior point. Then we study the boundary of the surface by analyzing limiting behavior of paths on the surface and sequences in a neighborhood of the surface. We introduce generic primal-dual infeasible-interior-point algorithms in which the search direction is the Newton direction for a system defining a point on the surface. We show that feasible-interior-point algorithms for artificial self-dual problems and for an artificial primal-dual pair of linear programming problems can he considered as special cases of these infeasible-interior-point algorithms or simple variants of them. In this sense, there are O(√nL)-iteration primal-dual infeasible-interior-point algorithms for linear programming.

The theory of linear programming:skew symmetric self-dual problems and the central path*

The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Guler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple approach which is based on an embedding of the primal problem and its dual into a skew symmetric self-dual problem. This embedding is essentially due Ye, Todd and Mizuno First we consider a skew symmetric self-dual linear program. We show that the strong duality theorem trivally holds in this case. Then, using the logarithmic barrier problem and the central path, the existence of a strictly complementary optimal solution is proved. Using the embedding just described, we easily obtain the strong duality theorem and the existence of strictly complementary optimal solutions for general linear programming problems

Symmetric duality with invexity in variational problems

Weak and strong duality results are established under invexity hypotheses for symmetric dual variational problems without positivity constraints. Self-dual problems and static symmetric dual programs are included as special cases.