Balinski and Tucker introduced in 1969 a special form of optimal tableaus for LP, which can be used to construct primal and dual optimal solutions such that the complementary slackness relation holds strictly. In this paper, first we note that using a polynomial time algorithm for LP Balinski and Tucker’s tableaus are obtainable in polynomial time. Furthermore, we show that, given a pair of pri...

In this paper we consider a general primal-dual nonlinear rescaling (PDNR) method for convex optimization with inequality constraints. We prove the global convergence of the PDNR method and estimate error bounds for the primal and dual sequences. In particular, we prove that, under the standard second-order optimality conditions the error bounds for the primal and dual sequences converge to zer...

This paper studies the asymptotic convergence properties of the primal-dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the ...

An adaptive choice for primal spaces, based on parallel sums, is developed for BDDC deluxe methods and elliptic problems in three dimensions. The primal space, which form the global, coarse part of the domain decomposition algorithm, and which is always required for any competitive algorithm, is defined in terms of generalized eigenvalue problems related to subdomain edges and faces; selected e...

Constrained Markov Decision Process (CMDP) is a natural framework for reinforcement learning tasks with safety constraints, where agents learn a policy that maximizes the long-term reward while satisfying the constraints on the long-term cost. A canonical approach for solving CMDPs is the primal-dual method which updates parameters in primal and dual spaces in turn. Existing methods for CMDPs o...

In this thesis, we study primal solutions for general optimization problems. In particular, we employ the subgradient method to solve the Lagrangian dual of a convex constrained problem, and use a primal-averaging scheme to obtain near-optimal and near-feasible primal solutions. We numerically evaluate the performance of the scheme in the framework of Network Utility Maximization (NUM), which h...

This paper characterizes the asymptotic convergence properties of the primal-dual dynamics to the solutions of a constrained concave optimization problem using classical notions from stability analysis. We motivate our study by providing an example which rules out the possibility of employing the invariance principle for hybrid automata to analyze the asymptotic convergence. We understand the s...

The Lagrange dual function is: g(u, v) = min x L(x, u, v) The corresponding dual problem is: maxu,v g(u, v) subject to u ≥ 0 The Lagrange dual function can be viewd as a pointwise maximization of some affine functions so it is always concave. The dual problem is always convex even if the primal problem is not convex. For any primal problem and dual problem, the weak duality always holds: f∗ ≥ g...

This paper presents a parametric approach for duality in fuzzy multi-criteria and multi-constraint level linear programming (MCLP) which extends fuzzy linear programming approaches. First, the MC-simplex method is used to solve the crisp prima–dual MCLP pair and then, through these crisp formulations, separate membership functions are constructed for fuzzy primal and dual program by considering...

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadr...