We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned … Read more
In this paper we introduce a new barrier function $\sum\limits_{i=1}^n(2x_i-1)[\ln{x_i}-\ln(1-x_i)]$ to solve the following optimization problem: $\min\,\, f(x)$ subject to: $Ax=b;\;\;0\leq x\leq e$. We show that this function is a $(3/2)n$-self-concordant barrier on the hypercube $[0,1]^n$. We prove that the central path is well defined and that under an additional assumption on the objective function, … Read more
We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in Faybusovich (1997) depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang (1998), where a vast set of conclusions concerning continuous trajectories is shown for monotone … Read more
In this paper we present optimization algorithms for image restoration based on the total variation (TV) minimization framework of L. Rudin, S. Osher and E. Fatemi (ROF). Our approach formulates TV minimization as a second-order cone program which is then solved by interior-point algorithms that are efficient both in practice (using nested dissection and domain … Read more
Sensitivity analysis is one of the most nteresting and preoccupying areas in optimization. Many attempts are made to investigate the problem’s behavior when the input data changes. Usually variation occurs in the right hand side of the constraints and/or the objective function coefficients. Degeneracy of optimal solutions causes considerable difficulties in sensitivity analysis. In this … Read more
We analyze a primal-dual interior-point method for nonlinear programming. We prove the global convergence for a wide class of problems under the standard assumptions on the problem. Citation Technical Report ORFE-04-07, Department of ORFE, Princeton University, Princeton, NJ 08544 Article Download View Convergence analysis of a primal-dual interior-point method for nonlinear programming
The paper presents an algorithm for solving nonlinear programming problems. The algorithm is based on the combination of interior and exterior point methods. The latter is also known as the primal-dual nonlinear rescaling method. The paper shows that in certain cases when the interior point method fails to achieve the solution with the high level … Read more
Conic optimization is the problem of optimizing a linear function over an intersection of an affine linear manifold with the Cartesian product of convex cones. However, many real world conic models involves an intersection rather than the product of two or more cones. It is easy to deal with an intersection of one or more … Read more
We consider two notions for the representations of convex cones: $G$-representation and lifted-$G$-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones … Read more
Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J.Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximity for linear optimization (LO) problems. They have also extended the approach for LO to SDO. In … Read more