In this paper, we consider a primal-dual interior point method for solving nonlinear semidefinite programming problems. We propose primal-dual interior point methods based on the unscaled and scaled Newton methods, which correspond to the AHO, HRVW/KSH/M and NT search directions in linear SDP problems. We analyze local behavior of our proposed methods and show their local and superlinear conver...

A pair of Wolfe type non-differentiable second order symmetric primal and dual problems in mathematical programming is formulated. The weak and strong duality theorems are then established under second order F-convexity assumptions. Symmetric minimax mixed integer primal and dual problems are also investigated. 2002 Elsevier Science B.V. All rights reserved.

Motivated by various applications to computer vision, we consider the convex cost tension problem, which is the dual of the convex cost flow problem. In this paper, we first propose a primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated minimum cut problems. We show that the time complexity of the prim...

In this paper, we are concerned with nonlinear minimization problems with second order cone constraints. A primal-dual interior point method is proposed for solving the problems. We also propose a new primal-dual merit function by combining the barrier penalty function and the potential function within the framework of the line search strategy, and show the global convergence property of our me...

Recently a new least-squares primal-dual (LSPD) algorithm, that is impervious to degeneracy, has effectively been applied to solving linear programming problems by Barnes et al., 2002. In this paper, we show an application of LSPD to shortest path problems with nonnegative arc length is equivalent to the Dijkstra’s algorithm. We also compare the LSPD algorithm with the conventional primal-dual ...

We observe a curious property of dual versus primal-dual path-following interior-point methods when applied to unbounded linear or conic programming problems in dual form. While primal-dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness, dual methods can sometimes implicitly move away from the analytic center of the set of infe...

We define the primal and dual linear programming problems involving interval numbers as the way of traditional linear programming problems. We discuss the solution concepts of primal and dual linear programming problems involving interval numbers without converting them to classical linear programming problems. By introducing new arithmetic operations between interval numbers, we prove the weak...

Given a convex optimization problem and its dual, there are many possible firstorder algorithms. In this paper, we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method. This is done through convex duality and implies notably that for certain problems, such as for supervised machine learning problems with nonsmooth losses or problems ...

Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? ...

We investigate the First-Order Primal-Dual (FPD) algorithm of Chambolle and Pock [1] in connection with MAP inference for general discrete graphical models. We provide a tight analytical upper bound of the stepsize parameter as a function of the underlying graphical structure (number of states, graph connectivity) and thus insight into the dependency of the convergence rate on the problem struc...