Jean-Philippe Vial

Convex Nondifferentiable Optimization: a Survey Focussed on the Analytic Center Cutting Plane Method

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.

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra’s d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities).

Deriving robust counterparts of nonlinear uncertain inequalities

In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.

ACCPM — A library for convex optimization based on an analytic center cutting plane method☆

Hardware information: Any computer with C++ and FORTRAN 77 compilers. Software information: C++ and FORTRAN 77.

Multiple Cuts in the Analytic Center Cutting Plane Method

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 Dikins 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 Dikins ellipsoid. We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(plog (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 Newtons method applied to a self-concordant function of p variables. The convergence result of [Ye, Math. Programming, 78 (1997), pp. 85--104] holds here also: the algorithm stops after

Shallow, deep and very deep cuts in the analytic center cutting plane method

O^*(\frac{\bar p^2n^2}{\varepsilon^2})

On the convergence of an infeasible primal-dual interior-point method for convex programming

cutting planes have been generated, where

Design and Operations of Gas Transmission Networks

\bar p

Experimental behavior of an interior point cutting plane algorithm for convex programming: an application to geometric programming

is the maximum number of cuts generated at any given iteration.

On Improvements to the Analytic Center Cutting Plane Method

In this paper we show that the cut does not need to go through the query point: it can be deep or shallow. The primal framework leads to a simple analysis of the potential variation, which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step.