Primal-dual interior-point methods for PDE-constrained optimization

This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L ^p . It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ^∞-setting is analyzed, but also a more involved L ^q -analysis, q < ∞, is presented. In L ^∞, the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L ^q -setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The L ^q -analysis with smoothing step yields global linear and local superlinear convergence, whereas the L ^∞-analysis without smoothing step yields only global linear convergence.     

基于指数型核函数的线性规划原始对偶内点算法

给出线性规划原始对偶内点算法的一个单变量指数型核函数.首先研究了这个指数型核函数的性质以及其对应的障碍函数.其次,基于这个指数型核函数,设计了求解线性规划问题的原始对偶内点算法,得到了目前小步算法最好的理论迭代界.最后,通过数值算例比较了基于指数型核函数的原始对偶内点算法和基于对数型核函数的原始对偶内点算法的计算效果.     

Primal-dual row-action method for convex programming

We present a primal-dual row-action method for the minimization of a convex function subject to general convex constraints. Constraints are used one at a time, no changes are made in the constraint functions and their Jacobian matrix (thus, the row-action nature of the algorithm), and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations. The algorithm generates two sequences: one of them, called primal, converges to the solution of the problem; the other one, called dual, approximates a vector of optimal KKT multipliers for the problem. We prove convergence of the primal sequence for general convex constraints. In the case of linear constraints, we prove that the primal sequence converges at least linearly and obtain as a consequence the convergence of the dual sequence.The research of the first author was partially supported by CNPq Grant No. 301280/86.     

Primal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term

In this paper we propose primal-dual interior-point algorithms for semidefinite optimization problems based on a new kernel function with a trigonometric barrier term. We show that the iteration bounds are $O(\sqrt{n}\log(\frac{n}{\epsilon}))$ for small-update methods and $O(n^{\frac{3}{4}}\log(\frac{n}{\epsilon}))$ for large-update, respectively. The resulting bound is better than the classical kernel function. For small-update, the iteration complexity is the best known bound for such methods.     

An interior-point method for semi-infinite programming problems

This work examines the generalization of a certain interior-point method, namely the method of analytic centers, to semi-infinite linear programming problems. We define an analytic center for these problems and an appropriate norm to examine Newton's method for computing this center. A simple algorithm of order zero is constructed and a convergence proof for that algorithm is given. Finally, we describe a more practical implementation of a predictor-corrector method and give some numerical results. In particular we concentrate on practical integration rules that take care of the specific structure of the integrals.     

An interior-point method for generalized linear-fractional programming

We develop an interior-point polynomial-time algorithm for a generalized linear-fractional problem. The latter problem can be regarded as a nonpolyhedral extension of the usual linear-fractional programming; typical example (which is of interest for control theory) is the minimization of the generalized eigenvalue of a pair of symmetric matrices linearly depending on the decision variables.     

An affine-scaling interior-point CBB method for box-constrained optimization

We develop an affine-scaling algorithm for box-constrained optimization which has the property that each iterate is a scaled cyclic Barzilai–Borwein (CBB) gradient iterate that lies in the interior of the feasible set. Global convergence is established for a nonmonotone line search, while there is local R-linear convergence at a nondegenerate local minimizer where the second-order sufficient optimality conditions are satisfied. Numerical experiments show that the convergence speed is insensitive to problem conditioning. The algorithm is particularly well suited for image restoration problems which arise in positron emission tomography where the cost function can be infinite on the boundary of the feasible set. This material is based upon work supported by the National Science Foundation under Grants 0203270, 0619080, and 0620286.     

An interior-point method for fractional programs with convex constraints

We present an interior-point method for a class of fractional programs with convex constraints. The proposed algorithm converges at a polynomial rate, similarly as in the case of a convex problem, even though fractional programs are only pseudo-convex. Here, the rate of convergence is measured in terms of the area of two-dimensional convex setsC _k containing the origin and certain projections of the optimal points, and the area ofC _k is reduced by a constant factorc < 1 at each iteration. The factorc depends only on the self-concordance parameter of a barrier function associated with the feasible set. We present an outline of a practical implementation of the proposed method, and we report results of some preliminary numerical experiments.Corresponding author.     

An interior-point method for solving box-constrained underdetermined nonlinear systems

A method introduced recently by Bellavia, Macconi and Morini for solving square nonlinear systems with bounds is modified and extended to cope with the underdetermined case. The algorithm produces a sequence of interior iterates and is based on globalization techniques due to Coleman and Li. Global and local convergence results are proved and numerical experiments are presented.     

A first-order interior-point method for linearly constrained smooth optimization

We propose a first-order interior-point method for linearly constrained smooth optimization that unifies and extends first-order affine-scaling method and replicator dynamics method for standard quadratic programming. Global convergence and, in the case of quadratic program, (sub)linear convergence rate and iterate convergence results are derived. Numerical experience on simplex constrained problems with 1000 variables is reported.     

An interior-point method for large constrained discrete ill-posed problems

Ill-posed problems are numerically underdetermined. It is therefore often beneficial to impose known properties of the desired solution, such as nonnegativity, during the solution process. This paper proposes the use of an interior-point method in conjunction with truncated iteration for the solution of large-scale linear discrete ill-posed problems with box constraints. An estimate of the error in the data is assumed to be available. Numerical examples demonstrate the competitiveness of this approach.     

An interior-point method for multifractional programs with convex constraints

We present an interior-point method for a family of multi-fractional programs with convex constraints. The programs under consideration consist of minimizing the maximum of a finite number of linear fractions over some convex set. First, we present a simple shortstep algorithm for solving such multifractional programs, and we show that, under suitable assumptions, the convergence of the short-step algorithm is weakly polynomial in a sense specified below. Then, we describe a practical implementation of the proposed method, and we report results of numerical experiments with this algorithm. These results suggest that the proposed method is a viable alternative to the standard Dinkelbach-type algorithms for solving multifractional programs.The authors would like to thank Professor A. S. Nemirovsky for stimulating discussions via electronic mail. We are grateful to two anonymous referees for comments and suggestions that improved our paper.     

Full Nesterov-Todd step infeasible interior-point method for symmetric optimization

Euclidean Jordan algebras were proved more than a decade ago to be an indispensable tool in the unified study of interior-point methods. By using it, we generalize the full-Newton step infeasible interior-point method for linear optimization of Roos [Roos, C., 2006. A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM Journal on Optimization. 16 (4), 1110-1136 (electronic)] to symmetric optimization. This unifies the analysis for linear, second-order cone and semidefinite optimizations.     

An interior-point piecewise linear penalty method for nonlinear programming

We present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ? ₂-penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The ? ₂-penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1?C36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1?C36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter???, iterates in a small neighborhood (roughly within o(??)) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.     

Feasibility issues in a primal-dual interior-point method for linear programming

A new method for obtaining an initial feasible interior-point solution to a linear program is presented. This method avoids the use of a big-M, and is shown to work well on a standard set of test problems. Conditions are developed for obtaining a near-optimal solution that is feasible for an associated problem, and details of the computational testing are presented. Other issues related to obtaining and maintaining accurate feasible solutions to linear programs with an interior-point method are discussed. These issues are important to consider when solving problems that have no primal or dual interior-point feasible solutions.     

Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

Mathematical Programming - In this paper, we propose a primal-dual path following method for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints,...     

On implementing a primal-dual interior-point method for conic quadratic optimization

 Based on the work of the Nesterov and Todd on self-scaled cones an implementation of a primal-dual interior-point method for solving large-scale sparse conic quadratic optimization problems is presented. The main features of the implementation are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type predictor-corrector extension and sparse linear algebra to improve the computational efficiency. Finally, the implementation exploits fixed variables which naturally occurs in many conic quadratic optimization problems. This is a novel feature for our implementation. Computational results are also presented to document that the implementation can solve very large problems robustly and efficiently. Received: November 18, 2000 / Accepted: January 18, 2001 Published online: September 27, 2002 Key Words. conic optimization – interior-point methods – large-scale implementation     

A globally convergent primal-dual interior-point filter method for nonlinear programming

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters.The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step.Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.Mathematics Subject Classification (1991): 65K05, 90C06, 90C29, 90C30Support for this author was provided by CRPC grant CCR–9120008.Support for this author was provided by CRPC grant CCR–9120008.Support for this author was provided by Centro de Matemática da Universidade de Coimbra, by FCT under grant POCTI/35059/MAT/2000, by the European Union under grant IST-2000-26063, and by Fundaç\ ao Calouste Gulbenkian. The author would also like to thank the IBM T.J. Watson Research Center and the Institute for Mathematics and Its Applications for their local support.     

Infeasible interior-point method for symmetric optimization using a positive-asymptotic barrier

We propose a new primal-dual infeasible interior-point method for symmetric optimization by using Euclidean Jordan algebras. Different kinds of interior-point methods can be obtained by using search directions based on kernel functions. Some search directions can be also determined by applying an algebraic equivalent transformation on the centering equation of the central path. Using this method we introduce a new search direction, which can not be derived from a usual kernel function. For this reason, we use the new notion of positive-asymptotic kernel function which induces the class of corresponding barriers. In general, the main iterations of the infeasible interior-point methods are composed of one feasibility and several centering steps. We prove that in our algorithm it is enough to take only one centering step in a main iteration in order to obtain a well-defined algorithm. Moreover, we conclude that the algorithm finds solution in polynomial time and has the same complexity as the currently best known infeasible interior-point methods. Finally, we give some numerical results.     

Primal-dual algorithms for linear programming based on the logarithmic barrier method

In this paper, we deal with primal-dual interior point methods for solving the linear programming problem. We present a short-step and a long-step path-following primal-dual method and derive polynomial-time bounds for both methods. The iteration bounds are as usual in the existing literature, namely iterations for the short-step variant andO(nL) for the long-step variant. In the analysis of both variants, we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long-step method depends strongly on the fact that the usual search directions form a descent direction for the so-called primal-dual logarithmic barrier function.This work was supported by a research grant from Shell, by the Dutch Organization for Scientific Research (NWO) Grant 611-304-028, by the Hungarian National Research Foundation Grant OTKA-2116, and by the Swiss National Foundation for Scientific Research Grant 12-26434.89.