We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a self-contained convergence analysis, that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.

We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variance--covariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid. We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(p log(p + 1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix ---primal, dual or primal-dual--- that is used in the computations. The computation of the optimal direction uses Newton's method applied to a self-concordant function of p variab...

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 problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.

We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a p-dimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution set, is a compact set consists of piecewise algebraic surfaces. We prove that the analytic center is recovered after adding a p-dimensional cut in O(p log(p 1)) damped Newton's iteration. We also prove that the algorithm stops when the dimension of the accumulated block diagonal matrix cut reaches to the bound of O (p 2 m 3 =ffl 2 ), where p is the maximum dimension cut and ffl is radius of the largest ball contained in the solution set.

In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method applied to convex optimization problems, which may lead to weaker results than Kelley's cutting plane method. Improvements to the analytic center cutting plane method are suggested. 1 Introduction In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method, and propose improvements. Cutting plane algorithms are designed to solve general convex optimization problems. They assume that the only information available around the current iterate takes the form of cutting planes, either supporting hyperplanes to the epigraph of the objective function, or separating hyperplanes from the feasible set. The two types of hyperplanes jointly define a linear programming, polyhedral, relaxation of the original convex optimization problem. The key issue in designing a specific cutting plane algorithm is the choice of a point in the current poly...

form of these problems can be described as finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices. Assume that Γ is defined by an oracle, which for any given m × m symmetric positive semidefinite matrix ˆ Y either confirms that ˆ Y ∈ Γ or returns a cut, i.e., a symmetric matrix A such that Γ is in the half-space {Y: A • Y ≤ A • ˆ Y}. We study an analytic center cutting plane algorithm for this problem. At each iteration the algorithm computes an approximate analytic center of a working set defined by the cutting-plane system generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise the new cutting plane returned by the oracle is added into the system. As the number of iterations increases, the working set shrinks and the algorithm eventually finds a solution of the problem. All iterates generated by the algorithm are positive definite matrices. The algorithm has a worst case complexity of O ∗ (m 3 /ɛ 2) on the total number of cuts to be used, where ɛ is the maximum radius of a ball contained by Γ.

This paper deals with an algorithm which incorporates the interior point method into the Dantzig-Wolfe decomposition technique for solving large-scale linear programming problems. At each iteration, the algorithm performs one step of Newton's method to solve a subproblem, obtaining an approximate solution, which is then used to compute an approximate Newton direction to find a new vector of the Lagrange multipliers. We show that the algorithm is globally linearly convergent and has the polynomial-time complexity. Key Words: Large-scale linear programming, Interior point methods, Dantzig-Wolfe decomposition, Complexity. Abbreviated Title: Interior point methods with decomposition AMS(MOS) subject classifications: 90C05, 90C06, 90C60. 1. Introduction This paper presents and analyzes an algorithm which incorporates the interior point method into the Dantzig-Wolfe decomposition method. Our concern in this paper is to show the polynomial-time complexity of the algorithm. In order to explo...

Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the relaxation. Typically, these cutting plane methods can be developed so as to exhibit polynomial convergence. The volumetric cutting plane algorithm achieves the theoretical minimum number of calls to a separation oracle. Long-step versions of the algorithms for solving convex optimization problems are presented. 1

In this paper, we analyze a logarithmic barrier decomposition method for solving a semiinfinite linear programming problem. This method is in some respects similar to the column generation methods using analytic centers. Although the method was found to be very efficient in the recent computational studies, its theoretical convergence or complexity is still unknown except in the (finite) case of linear programming. In this paper we present a complexity analysis of this method in the general semi-infinite case. Our complexity estimate is given in terms of the problem dimension, the radius of the largest Euclidean ball contained in the feasible set, and the desired accuracy of the approximate solution. KEY WORDS. Semi-infinite linear programming, logarithmic barrier, decomposition, column generation. AMS subject classification: 90C25, 90C60. iii 1 Introduction Consider the following semi-infinite linear programming problem (SILP): maximize f 0 (y) := a T 0 y subject to f t (y) := ...

We propose a generic path-following scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic procedure in use, primal, dual or primal-dual. We show convergence in O( p n) iterations. We verify that the primal, dual and primal-dual algorithms satisfy the local quadratic convergence property. The method can be applied to solve the linear programming problem (with a feasible start) and to compute the analytic center of a bounded polytope. The generic path-following scheme easily extends to the logarithmic penalty barrier approach. Keywords Interior point method, method of centers, path-following, linear programming. This work has been completed with the support from the Swiss National Foundation for Scientific Research, grant 12-34002.92. 1 Introduction Shortly after Karmarkar's s...