Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to d...

We present combinatorial interior point methods for the generalized minimum cost flow and the generalized circulation problems based on Wallacher and Zimmermann’s combinatorial interior point method for the minimum cost network flow problem. The algorithms have features of both a combinatorial algorithm and an interior point method. They work towards optimality by iteratively reducing the value...

Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow probl...

In this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as the XS+SXmethod, this clas...

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 ∇ and ∇∗ which roughly corresponds to the primal and the dual problem. The dua...

We propose a primal-dual interior-point algorithm for semidefinite optimization(SDO) based on a class of kernel functions which are both eligible and self-regular. New search directions and proximity measures are defined based on these functions. We show that the algorithm has O( √ n log ε ) and O( √ n logn log ε ) complexity results for smalland large-update methods, respectively. These are th...

We propose a family of search directions based on primal-dual entropy in the contextof interior-point methods for linear optimization. We show that by using entropy based searchdirections in the predictor step of a predictor-corrector algorithm together with a homogeneousself-dual embedding, we can achieve the current best iteration complexity bound for linear opti-mization. The...

In this paper we analyze inexact trust–region interior–point (TRIP) sequential quadra– tic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonlinear constr...

A linear program is typically specified by a matrix A together with two vectors b and c, where A is an n-by-d matrix, b is an n-vector and c is a d-vector. There are several canonical forms for defining a linear program using (A, b, c). One commonly used canonical form is: max cx s.t. Ax ≤ b and its dual min by s.t A y = c, y ≥ 0. In [Ren95b, Ren95a, Ren94], Renegar defined the condition number...

Linear Complementarity Problems (LCPs) belong to the class of NP-complete problems. Therefore we can not expect a polynomial time solution method for LCPs without requiring some special property of the coefficient matrix. Following our recently published ideas we generalize affine scaling and predictor-corrector interior point algorithms to solve LCPs with general matrices in EP-sense, namely, ...