Computational experience with an interior point algorithm on the satisfiability problem

We apply the zero-one integer programming algorithm described in Karmarkar [12] and Karmarkar, Resende and Ramakrishnan [13] to solve randomly generated instances of the satisfiability problem (SAT). The interior point algorithm is briefly reviewed and shown to be easily adapted to solve large instances of SAT. Hundreds of instances of SAT (having from 100 to 1000 variables and 100 to 32,000 clauses) are randomly generated and solved. For comparison, we attempt to solve the problems via linear programming relaxation with MINOS.     

Using a facility location algorithm to solve large set covering problems

Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.     

A note on some computationally difficult set covering problems

Fulkerson et al. have given two examples of set covering problems that are empirically difficult to solve. They arise from Steiner triple systems and the larger problem, which has a constraint matrix of size 330 × 45 has only recently been solved. In this note, we show that the Steiner triple systems do indeed give rise to a series of problems that are probably hard to solve by implicit enumeration. The main result is that for ann variable problem, branch and bound algorithms using a linear programming relaxation, and/or elimination by dominance require the examination of a super-polynomial number of partial solutionsThis paper was written while the author was a CORE Fellow at the Université de Louvain, Louvain-la-Neuve, Belgium.     

An active-set strategy in an interior point method for linear programming

We will present a potential reduction method for linear programming where only the constraints with relatively small dual slacks—termed active constraints—will be taken into account to form the ellipsoid constraint at each iteration of the process. The algorithm converges to the optimal feasible solution in O( L) iterations with the same polynomial bound as in the full constraints case, wheren is the number of variables andL is the data length. If a small portion of the constraints is active near the optimal solution, the computational cost to find the next direction of movement in one iteration may be considerably reduced by the proposed strategy.This research was partially done in June 1990 while the author was visiting the Department of Mathematics, University of Pisa.     

Genetic algorithms applied to computationally difficult set covering problems

The set covering problem is a known NP-hard combinatorial optimisation problem for covering the rows of a matrix by a subset of columns at minimum cost. Genetic algorithms (GA) are a class of iteration procedures that simulate the evolution process of a structured population. The objective of this work is to show that a somewhat classical GA implementation reaches high quality computational results for difficult set covering problems arising in computing the 1-width of incidence matrices of Steiner triple systems. In computational tests all optimal and best known solutions were found for incidence matrices A₉, A₁₅, A₂₇, A₄₅, A₈₁ and A₂₄₃ with reasonable times for a microcomputer implementation. Other tests with classical set covering problems confirm the good results for an additional class of instances.     

An O(n 3 L) primal interior point algorithm for convex quadratic programming

We present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach. This approach generates a sequence of problems, each of which is approximately solved by taking a single Newton step. It is shown that the method requires iterations and O(n ^3.5 L) arithmetic operations. By using modified Newton steps the number of arithmetic operations required by the algorithm can be reduced to O(n ³ L).This research was supported in part by NSF Grant DMS-85-12277 and ONR Contract N-00014-87-K0214. It was presented at the Meeting on Mathematische Optimierung, Mathematisches Forschungsinstitut, Oberwolfach, West Germany, January 3–9, 1988.     

An algorithm to solve polyhedral convex set optimization problems

An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain satisfying two conditions: the attainment of the infimum and minimality with respect to a set relation. In the first phase of the algorithm, a linear vector optimization problem, called the vectorial relaxation, is solved. The resulting pre-solution yields the attainment of the infimum but, in general, not minimality. In the second phase of the algorithm, minimality is established by solving certain linear programs in combination with vertex enumeration of some values of the objective map.     

An extension of interior point potential reduction algorithm to solve general LCPs

Linear complementarity problems (LCP) may be solved if their condition number, defined appropriately is positive. This paper presents a transformation for the LCP which satisfy mild conditions to a form which will have a positive condition number. For these extensive classes an Interior point method can be used to solve it.     

An interior point method for general large-scale quadratic programming problems

In this paper, we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems efficiently solves a wide class of largescale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a three-dimensional cost improvement subproblem, which is solved at every interation. We have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the sucess of the algorithm. We describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented.Contribution of the National Institute of Standards and Tedchnology and not subject to copyright in the United States. This research was supported in part by ONR Contract N-0014-87-F0053.     

Note: An algorithm to solve polyhedral convex set optimization problems

We provide a counterexample to the remark in Löhne and Schrage [An algorithm to solve polyhedral convex set optimization problems, Optimization 62 (2013), pp. 131-141] that every solution of a polyhedral convex set optimization problem is a pre-solution. A correct statement is that every solution of a polyhedral convex set optimization problem obtained by the algorithm SetOpt is a pre-solution. We also show that every finite infimizer and hence every solution of a polyhedral convex set optimization problem contains a pre-solution.     

A combinatorial interior point method for network flow problems

For solving minimum cost flow problems, we develop a combinatorial interior point method based on a variant of the algorithm of Karmarkar, described in Gonzaga [3, 4]. Gonzaga proposes search directions generated by projecting certain directions onto the nullspace ofA. By the special combinatorial structure of networks any projection onto the nullspace ofA can be interpreted as a flow in the incremental graph ofG. In particular, to evaluate the new search direction, it is sufficient to choose a negative circuit subject to costs on the arcs depending on the current solution. That approach results in an O(mn ² L) algorithm wherem denotes the number of vertices,n denotes the number of arcs, andL denotes the total length of the input data.     

Polynomiality of an inexact infeasible interior point algorithm for semidefinite programming

In this paper we present a primal-dual inexact infeasible interior-point algorithm for semidefinite programming problems (SDP). This algorithm allows the use of search directions that are calculated from the defining linear system with only moderate accuracy, and does not require feasibility to be maintained even if the initial iterate happened to be a feasible solution of the problem. Under a mild assumption on the inexactness, we show that the algorithm can find an -approximate solution of an SDP in O(n²ln(1/)) iterations. This bound of our algorithm is the same as that of the exact infeasible interior point algorithms proposed by Y. Zhang.Research supported in part by the Singapore-MIT alliance, and NUS Academic Research Grant R-146-000-032-112.Mathematics Subject Classification (1991): 90C05, 90C30, 65K05     

Interior proximal point algorithm for linear programs

An interior proximal point algorithm for finding a solution of a linear program is presented. The distinguishing feature of this algorithm is the addition of a quadratic proximal term to the linear objective function. This perturbation has allowed us to obtain solutions with better feasibility. Implementation of this algorithm shows that the algorithms. We also establish global convergence and local linear convergence of the algorithm.This research was supported by National Science Foundation Grants DCR-85-21228 and CCR-87-23091 and by Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-89-0410. It was conducted while the author was a Graduate Student at the Computer Sciences Department, University of Wisconsin, Madison, Wisconsin.     

An interior point method for quadratic programs based on conjugate projected gradients

We propose an interior point method for large-scale convex quadratic programming where no assumptions are made about the sparsity structure of the quadratic coefficient matrixQ. The interior point method we describe is a doubly iterative algorithm that invokes aconjugate projected gradient procedure to obtain the search direction. The effect is thatQ appears in a conjugate direction routine rather than in a matrix factorization. By doing this, the matrices to be factored have the same nonzero structure as those in linear programming. Further, one variant of this method istheoretically convergent with onlyone matrix factorization throughout the procedure.     

Projected Hessian algorithm with backtracking interior point technique for linear constrained optimization

In this paper, we propose a new trust-region-projected Hessian algorithm with nonmonotonic backtracking interior point technique for linear constrained optimization. By performing the QR decomposition of an affine scaling equality constraint matrix, the conducted subproblem in the algorithm is changed into the general trust-region subproblem defined by minimizing a quadratic function subject only to an ellipsoidal constraint. By using both the trust-region strategy and the line-search technique, each iteration switches to a backtracking interior point step generated by the trustregion subproblem. The global convergence and fast local convergence rates for the proposed algorithm are established under some reasonable assumptions. A nonmonotonic criterion is used to speed up the convergence in some ill-conditioned cases. Selected from Journal of Shanghai Normal University (Natural Science), 2003, 32(4): 7–13     

A note on a symmetrical set covering problem: The lottery problem

In a lottery, n numbers are drawn from a set of m numbers. On a lottery ticket we fill out n numbers. Consider the following problem: what is the minimum number of tickets so that there is at least one ticket with at least p matching numbers? We provide a set-covering formulation for this problem and characterize its LP solution. The existence of many symmetrical alternative solutions, makes this a very difficult problem to solve, as our computational results indicate.     

A boundary perturbation interior point homotopy method for solving fixed point problems

In this paper, a boundary perturbation interior point homotopy method is proposed to give a constructive proof of the general Brouwer fixed point theorem and thus solve fixed point problems in a class of nonconvex sets. Compared with the previous results, by using the newly proposed method, initial points can be chosen in the whole space of Rn, which may improve greatly the computational efficiency of reduced predictor-corrector algorithms resulted from that method. Some numerical examples are given to illustrate the results of this paper.     

Extension of primal-dual interior point methods to diff-convex problems on symmetric cones

We consider the extension of primal dual interior point methods for linear programming on symmetric cones, to a wider class of problems that includes approximate necessary optimality conditions for functions expressible as the difference of two convex functions of a special form. Our analysis applies the Jordan-algebraic approach to symmetric cones. As the basic method is local, we apply the idea of the filter method for a globalization strategy.     

A primal interior point method for the linear semidefinite programming problem

The linear semidefinite programming problem is examined. A primal interior point method is proposed to solve this problem. It extends the barrier-projection method used for linear programs. The basic properties of the proposed method are discussed, and its local convergence is proved.