An exact penalty function for semi-infinite programming

This paper introduces a global approach to the semi-infinite programming problem that is based upon a generalisation of the ℓ₁ exact penalty function. The advantages are that the ensuing penalty function is exact and the penalties include all violations. The merit function requires integrals for the penalties, which provides a consistent model for the algorithm. The discretization is a result of the approximate quadrature rather than an a priori aspect of the model. This research was partially supported by Natural Sciences and Engineering Research Council of Canada grants A-8639 and A-8442. This paper was typeset using software developed at Bell Laboratories and the University of California at Berkeley.     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

An exact algorithm for graph partitioning

An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate. It is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set. The concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function Hessian, or by a diagonal shift obtained by solving a semidefinite programming problem. Numerical results show that the proposed algorithm is competitive with state-of-the-art graph partitioning codes.     

Recursive quadratic programming algorithm that uses an exact augmented Lagrangian function

An algorithm for nonlinear programming problems with equality constraints is presented which is globally and superlinearly convergent. The algorithm employs a recursive quadratic programming scheme to obtain a search direction and uses a differentiable exact augmented Lagrangian as line search function to determine the steplength along this direction. It incorporates an automatic adjustment rule for the selection of the penalty parameter and avoids the need to evaluate second-order derivatives of the problem functions. Some numerical results are reported.     

An exact penalty function for nonlinear programming with inequalities

It is shown how, given a nonlinear programming problem with inequality constraints, it is possible to construct an exact penalty function with a local unconstrained minimum at any local minimum of the constrained problem. The unconstrained minimum is sufficiently smooth to permit conventional optimization techniques to be used to locate it. Numerical evidence is presented on five well-known test problems.     

A recursive quadratic programming algorithm that uses differentiable exact penalty functions

In this paper, a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied. The line search functions used are approximations to Fletcher's differentiable exact penalty function. Global convergence and local superlinear convergence results are proved, and some numerical results are given.     

An exact algorithm for team orienteering problems

Optimising routing of vehicles constitutes a major logistic stake in many industrial contexts. We are interested here in the optimal resolution of special cases of vehicle routing problems, known as team orienteering problems. In these problems, vehicles are guided by a reward that can be collected from customers, while the length of routes is limited. The main difference with classical vehicle routing problems is that not all customers have to be visited. The solution method we propose here is based on a Branch & Price algorithm. It is, as far as we know, the first exact method proposed for such problems, except for a preliminary work from Gueguen (Methodes de résolution exacte pour problémes de tournées de véhicules. Thése de doctorat, école Centrale Paris 1999) and a work from Butt and Ryan (Comput Oper Res 26(4):427–441 1999). It permits to solve instances with up to 100 customers.      

Bilinear programming and structured stochastic games

One-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers. Solutions to such classes of games under the average reward criterion can be derived from optimal solutions to appropriate bilinear programs. The validity of using bilinear programming as a solution method follows from two preliminary theorems, the first of which is a complete classification of undiscounted stochastic games with optimal stationary strategies. The second of these preliminary theorems relaxes the conditions of the classification theorem for certain classes of stochastic games and provides the basis for the bilinear programming results. Analogous results hold for the discounted stochastic games with the above special structures.This research was supported in part by the Air Force Office of Scientific Research and by the National Science Foundation under Grant No. ECS-850-3440.     

An efficient sequential quadratic programming algorithm for nonlinear programming

In this paper, a new feasible sequential quadratic programming (FSQP) algorithm is proposed to solve the nonlinear programming, where a feasible descent direction is obtained by solving only one QP subproblem. In order to avoid Maratos effect, a high-order revised direction is computed by solving a linear system with involving some “active” constraints. The theoretical analysis shows that global and superlinear convergence can be deduced.     

An exact penalty global optimization approach for mixed-integer programming problems

In this work, we propose a global optimization approach for mixed-integer programming problems. To this aim, we preliminarily define an exact penalty algorithm model for globally solving general problems and we show its convergence properties. Then, we describe a particular version of the algorithm that solves mixed-integer problems and we report computational results on some MINLP problems.     

An exact penalty method for weak linear bilevel programming problem

In this paper, we present an exact penalty method, which is different from the existing penalty method, for solving weak linear bilevel programming problem. Then, we establish an existence result of solutions for such a problem. Finally, we propose an algorithm and give two examples to illustrate its feasibility.     

An exact penalty function approach for nonlinear integer programming problems

Nonlinear integer programming problems with bounded feasible sets are considered. It is shown how the number of constraints in such problems can be reduced with the aid of an exact penalty function approach. This approach can be used to construct an equivalent unconstrained problem, or a problem with a constraint set which makes it easier to solve. The application of this approach to various nonlinear integer programming problems is discussed.     

An efficient algorithm for linear programming

A simple but efficient algorithm is presented for linear programming. The algorithm computes the projection matrix exactly once throughout the computation unlike that of Karmarkar’s algorithm where in the projection matrix is computed at each and every iteration. The algorithm is best suitable to be implemented on a parallel architecture. Complexity of the algorithm is being studied.     

An ellipsoid algorithm for nonlinear programming

We investigate an ellipsoid algorithm for nonlinear programming. After describing the basic steps of the algorithm, we discuss its computer implementation and present a method for measuring computational efficiency. The computational results obtained from experimenting with the algorithm are discussed and the algorithm's performance is compared with that of a widely used commercial code. This research was supported in part by The National Science Foundation, Grant No. MCS78-02096.     

An algorithm for bicriterion linear programming

An algorithm is presented here for bicriterion linear programs which generates either all efficient points or a subset of such efficient points corresponding to a decision-maker's specified space of objective weights. The computational requirements of the algorithm are quite low; in fact only a series of divisions and comparisons are needed for the determination of adjacent efficient extreme points. As a by-product, the range of objective weights corresponding to each efficient extreme point is also generated. This additional information is used to characterize the set of all efficient points as a union of maximal efficient faces.     

An algorithm for nonconvex programming problems

Branch and bound approaches for nonconvex programming problems had been given in [1] and [4]. Crucial for both are the use of rectangular partitions, convex envelopes and separable nonconvex portions of the objective function and constraints. We want to propose a similar algorithm which solves a sequence of problems in each of which the objective function is convex or even linear. The main difference between this approach and previous approaches is the use of general compact partitions instead of rectangular ones and a different refining rule such that the algorithm does not rely on the concept of convex envelopes and handles non-separable functions.First we describe a general algorithm and prove a convergence theorem under suitable regularity assumptions. Then we give as example an algorithm for concave minimization problems.     

An exact algorithm for the network pricing problem

This work focuses on an improved exact algorithm for addressing an NP-hard network pricing problem. The method involves an efficient and partial generation of candidate solutions, a recursive scheme for generating improved upper bounds, and a column generation procedure for solving the network-structured subproblems. Its efficiency is assessed against both randomly generated instances involving three distinct topologies as well as instances based on real life situations in telecommunication and freight transportation.     

An exact algorithm for the fuzzy p-median problem

In this paper we propose a fuzzy version of the classical p-median problem. We consider a fuzzy set of constraints so that the decision-maker will be able to take into account solutions which provide significantly lower costs by leaving a part of the demand uncovered. We propose an algorithm for solving the problem which is based on Hakimi's works and we compare the crisp and the fuzzy approach by means of an example.     

An improved bit parallel exact maximum clique algorithm

This paper describes new improvements for BB-MaxClique (San Segundo et al. in Comput Oper Resour 38(2):571–581, 2011), a leading maximum clique algorithm which uses bit strings to efficiently compute basic operations during search by bit masking. Improvements include a recently described recoloring strategy in Tomita et al. (Proceedings of the 4th International Workshop on Algorithms and Computation. Lecture Notes in Computer Science, vol 5942. Springer, Berlin, pp 191–203, 2010), which is now integrated in the bit string framework, as well as different optimization strategies for fast bit scanning. Reported results over DIMACS and random graphs show that the new variants improve over previous BB-MaxClique for a vast majority of cases. It is also established that recoloring is mainly useful for graphs with high densities.