We consider the logarithmic and the volumetric barrier functions used in interior point methods. In the case of the logarithmic barrier function, the analytic center of a level set is the point at which the central path intersects that level set. We prove that this also holds for the volumetric path. For the central path, … Read more
In this paper we study interior point (IP) methods for solving optimal design problems. In particular, we propose a primal IP method for solving the problems with general convex optimality criteria and establish its global convergence. In addition, we reformulate the problems with A-, D- and E-criterion into linear or log-determinant semidefinite programs (SDPs) and … Read more
We study the interactions between computational and economic performance of dispatch operations under highly dynamic environments. In particular, we discuss the need for extending the forecast horizon of the dispatch formulation in order to anticipate steep variations of renewable power and highly elastic loads. We present computational strategies to solve the increasingly larger optimization problems … Read more
The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Most of studies done on the SCCP can be categorized into the three research themes, interior point methods for the SCCP, merit or smoothing function methods for … Read more
Penalty and interior-point methods for nonlinear optimization problems have enjoyed great successes for decades. Penalty methods have proved to be effective for a variety of problem classes due to their regularization effects on the constraints. They have also been shown to allow for rapid infeasibility detection. Interior-point methods have become the workhorse in large-scale optimization … Read more
We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of … Read more
In this paper, we study polynomial-time interior-point algorithms in view of information geometry. We introduce an information geometric structure for a conic linear program based on a self-concordant barrier function. Riemannian metric is defined with the Hessian of the barrier function. We introduce two connections $\nabla$ and $\nabla^*$ which roughly corresponds to the primal and … Read more
In this paper we propose a warm-start technique for interior point methods applicable to multi-stage stochastic programming problems. The main idea is to generate an initial point for the interior point solver by decomposing the barrier problem associated with the deterministic equivalent at the sec- ond stage and using a concatenation of the solutions of … Read more
We consider the problem to minimize the sum of piecewise-linear convex functions under both linear and nonnegative constraints. We convert the piecewise-linear convex problem into a standard form linear programming problem (LP) and apply a primal-dual interior-point method for the LP. From the solution of the converted problem, we can obtain the solution of the … Read more
In this work we consider the optimal control problem of a semilinear elliptic PDE with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated parametrized family of penalized problems, whose solutions define a central path converging to the … Read more