The primal-dual algorithm for linear programming is very effective for solving network flow problems. For the method to work, an initial feasible solution to the dual is required. In this article, we show that, for the shortest path problem in a positively weighted graph equipped with a consistent heuristic function, the primal-dual algorithm will become the well-known A* algorithm if a special...

In this paper we propose distributed dual gradient algorithms for linearly constrained separable convex problems and analyze their rate of convergence under different assumptions. Under the strong convexity assumption on the primal objective function we propose two distributed dual fast gradient schemes for which we prove sublinear rate of convergence for dual suboptimality but also primal subo...

In this paper we will deal with primal{dual interior point methods for solving the linear programming problem. We present a short step and a long step path{following primal{dual method and derive polynomial{ time bounds for both methods. The iteration bounds are as usual in the existing literature, namely O(pnL) iterations for the short step, and O(nL) for the long step variant. In the analysis...

In this paper we propose a new large-update primal-dual interior point algorithm for P∗( ) linear complementarity problems (LCPs). We generalize Bai et al.’s [A primal-dual interior-point method for linear optimization based on a new proximity function, Optim. Methods Software 17(2002) 985–1008] primal-dual interior point algorithm for linear optimization (LO) problem to P∗( ) LCPs. New search ...

We establish connections between the facial reduction algorithm of Borwein and Wolkowicz and the self-dual homogeneous model of Goldman and Tucker when applied to conic optimization problems. Specifically, we show the self-dual homogeneous model returns facial reduction certificates when it fails to return a primal-dual optimal solution or a certificate of infeasibility. Using this observation,...

The primal-dual Dikin-type affine scaling method was originally proposed for linear optimization and then extended to semidefinite optimization. Here, the method is generalized to symmetric conic optimization using the notion of Euclidean Jordan algebras. The method starts with an interior feasible but not necessarily centered primal-dual solution, and it features both centering and reducing th...

The Pseudo Primal-Dual Algorithm solves the pure integer programming problem in two stages, systematically violating and restoring dual feasibility while maintaining an all-integer matrix. The algorithm is related to Gomory AlI-fnteger Algorithm and the Young Primal Integer Programming Algorithm, differing from the former in the dual feasible stage by the choice of cuts and pivot variable, and ...

We consider a primal-dual algorithm for minimizing f(x) + h(Ax) with differentiable f . The primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFPO) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we extend it to solve f(x) + h l(Ax) with differentiable l and prove its convergence under a weak condition (i.e...

This thesis aims to introduce and analyse a primal-dual active set strategy for solving Allen-Cahn variational inequalities. We consider the standard Allen-Cahn equation with non-local constraints and a vector-valued Allen-Cahn equation with and without non-local constraints. Existence and uniqueness results are derived in a formulation involving Lagrange multipliers for local and non-local con...

In this paper we analyze the rate of local convergence of the Newton primal-dual interiorpoint method when the iterates are kept strictly feasible with respect to the inequality constraints. It is shown under the classical conditions that the rate is q–quadratic when the functions associated to the binding inequality constraints are concave. In general, the q–quadratic rate is achieved provided...