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.     

A noninterior point homotopy method for semi-infinite programming problems

In this paper, we present a new homotopy method which is a non-interior point homotopy method for solving semi-infinite programming problems. Under suitable assumptions, we prove that the method determines a smooth path from a given point. The new homotopy method generalizes the existing combined homotopy interior point method for semi-infinite programming problems to unbounded set, moreover, it is more convenient in that it enlarges the choice scope of the initial point. Some numerical examples are given to show its efficiency.     

A homotopy interior point method for semi-infinite programming problems

This paper presents a homotopy interior point method for solving a semi-infinite programming (SIP) problem. For algorithmic purpose, based on bilevel strategy, first we illustrate appropriate necessary conditions for a solution in the framework of standard nonlinear programming (NLP), which can be solved by homotopy method. Under suitable assumptions, we can prove that the method determines a smooth interior path from a given interior point to a point w ^*, at which the necessary conditions are satisfied. Numerical tracing this path gives a globally convergent algorithm for the SIP. Lastly, several preliminary computational results illustrating the method are given.     

Application of general semi-infinite programming to lapidary cutting problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interior-point method developed by Stein [O. Stein, Bi-level Strategies in Semi-infinite Programming, Kluwer Academic Publishers, Boston, 2003]. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on real-world data are also presented.     

Duality for nonsmooth semi-infinite programming problems

This paper deals with nonsmooth semi-infinite programming problem which in recent years has become an important field of active research in mathematical programming. A semi-infinite programming problem is characterized by an infinite number of inequality constraints. We formulate Wolfe as well as Mond-Weir type duals for the nonsmooth semi-infinite programming problem and establish weak, strong and strict converse duality theorems relating the problem and the dual problems. To the best of our knowledge such results have not been done till now.     

An inexact approach to solving linear semi-infinite programming problems

In this paper, we propose an “inexact solution” approach to deal with linear semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. A general convergence proof with some numerical examples are given and the advantages of using this approach are discussed     

An interior semi-infinite programming method

In this paper, a new method for semi-infinite programming problems with convex constraints is presented. The method generates a sequence of feasible points whose cluster points are solutions of the original problem. No maximization over the index set is required. Some computational results are also presented.This work was partly supported by Republicka Zajednica za Nauku Socijalisticke Republike Srbije. The authors are indebted to Professor R. A. Tapia for encouraging the approach taken in this research.     

Understanding linear semi-infinite programming via linear programming over cones

In this article, the standard primal and dual linear semi-infinite programming (DLSIP) problems are reformulated as linear programming (LP) problems over cones. Therefore, the dual formulation via the minimal cone approach, which results in zero duality gap for the primal–dual pair for LP problems over cones, can be applied to linear semi-infinite programming (LSIP) problems. Results on the geometry of the set of the feasible solutions for the primal LSIP problem and the optimality criteria for the DLSIP problem are also discussed.     

Nonsmooth semi-infinite programming problems with mixed constraints

We consider a nonsmooth semi-infinite programming problem with a feasible set defined by inequality and equality constraints and a set constraint. First, we study some alternative theorems which involve linear and sublinear functions and a convex set and we propose several generalizations of them. Then, alternative theorems are applied to obtain, under different constraint qualifications, several necessary optimality conditions in the type of Fritz-John and Karush-Kuhn-Tucker.     

A nonsmooth Levenberg–Marquardt method for solving semi-infinite programming problems

In this paper, we first transform the semi-infinite programming problem into the KKT system by the techniques in [D.H. Li, L. Qi, J. Tam, S.Y. Wu, A smoothing Newton method for semi-infinite programming, J. Global. Optim. 30 (2004) 169–194; L. Qi, S.Y. Wu, G.L. Zhou, Semismooth Newton methods for solving semi-infinite programming problems, J. Global. Optim. 27 (2003) 215–232]. Then a nonsmooth and inexact Levenberg–Marquardt method is proposed for solving this KKT system based on [H. Dan, N. Yamashita, M. Fukushima, Convergence properties of the inexact Levenberg–Marquardt method under local error bound conditions, Optimim. Methods Softw., 11 (2002) 605–626]. This method is globally and superlinearly (even quadratically) convergent. Finally, some numerical results are given.     

Discretization methods for the solution of semi-infinite programming problems

In the first part of this paper, we prove the convergence of a class of discretization methods for the solution of nonlinear semi-infinite programming problems, which includes known methods for linear problems as special cases. In the second part, we modify and study this type of algorithms for linear problems and suggest a specific method which requires the solution of a quadratic programming problem at each iteration. With this algorithm, satisfactory results can also be obtained for a number of singular problems. We demonstrate the performance of the algorithm by several numerical examples of multivariate Chebyshev approximation problems.     

Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

In this paper, we present an algorithm to solve nonlinear semi-infinite programming (NSIP) problems. To deal with the nonlinear constraint, Floudas and Stein (SIAM J. Optim. 18:1187?C1208, 2007) suggest an adaptive convexification relaxation to approximate the nonlinear constraint function. The ??BB method, used widely in global optimization, is applied to construct the convexification relaxation. We then combine the idea of the cutting plane method with the convexification relaxation to propose a new algorithm to solve NSIP problems. With some given tolerances, our algorithm terminates in a finite number of iterations and obtains an approximate stationary point of the NSIP problems. In addition, some NSIP application examples are implemented by the method proposed in this paper, such as the proportional-integral-derivative controller design problem and the nonlinear finite impulse response filter design problem. Based on our numerical experience, we demonstrate that our algorithm enhances the computational speed for solving NSIP problems.     

A norm-relaxed method of feasible directions for finely discretized problems from semi-infinite programming

In this paper, a class of finely discretized Semi-Infinite Programming (SIP) problems is discussed. Combining the idea of the norm-relaxed Method of Feasible Directions (MFD) and the technique of updating discretization index set, we present a new algorithm for solving the Discretized Semi-Infinite (DSI) problems from SIP. At each iteration, the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem, i.e., a quadratic program, and some appropriate constraints are chosen to reduce the computational cost. A high-order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints. Under weak conditions such as Mangasarian–Fromovitz Constraint Qualification (MFCQ), the proposed algorithm possesses weak global convergence. Moreover, the superlinear convergence is obtained under Linearly Independent Constraint Qualification (LICQ) and other assumptions. In the end, some elementary numerical experiments are reported.     

Necessary optimality conditions for nonsmooth semi-infinite programming problems

In this paper, for a nonsmooth semi-infinite programming problem where the objective and constraint functions are locally Lipschitz, analogues of the Guignard, Kuhn-Tucker, and Cottle constraint qualifications are given. Pshenichnyi-Levin-Valadire property is introduced, and Karush-Kuhn-Tucker type necessary optimality conditions are derived.     

Hidden conic quadratic representation of some nonconvex quadratic optimization problems

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.     

Second order optimality conditions for generalized semi-infinite programming problems

We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditionsfor local optimality are given. The conditions are derived under assumptions such that the feasible set can be described by means of a finite number of optimal value functions. Since we do not require a strict complementary condition for the local reduction these functions are only of class C¹ A sufficient condition for optimality is proven under much weaker assumptions.     

A recursive quadratic programming algorithm for semi-infinite optimization problems

The well known, local recursive quadratic programming method introduced by E. R. Wilson is extended to apply to optimization problems with constraints of the type , where ranges over a compact interval of the real line. A scheme is proposed, which results in a globally convergent conceptual algorithm. Finally, two implementable versions are presented both of which converge quadratically.Research sponsored by the National Science Foundation Grant ECS-79-13148 and the Air Force Office of Scientific Research (AFOSR) United States Air Force Contract No. F49620-79-C-0178     

Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems

This paper is devoted to the study of nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. We introduce a constraint qualification which is based on the Mordukhovich subdifferential. Then, we derive a Fritz–John type necessary optimality condition. Finally, interrelations between the new and the existing constraint qualifications such as the Mangasarian–Fromovitz, linear independent, and the Slater are investigated.     

Sensitivity method for basis inverse representation in multistage stochastic linear programming problems

A version of the simplex method for solving stochastic linear control problems is presented. The method uses a compact basis inverse representation that extensively exploits the original problem data and takes advantage of the supersparse structure of the problem. Computational experience indicates that the method is capable of solving large problems.This research was supported by Programs CPBP02.15 and RPI.02.