Finding normal solutions in piecewise linear programming

Letf: ⁿ (–, ] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min _x ¦x¦²/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} _t 0, minimizingf over an ellipsoid, and choosing penalty parameters inL ₁QP methods for strictly convex QP.This research was supported by the State Committee for Scientific Research under Grant 8S50502206.     

Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint

This paper extends and completes the discussion by Xing et?al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted) about the quadratic programming over one quadratic constraint (QP1QC). In particular, we relax the assumption to cover more general cases when the two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence. Under such an assumption, the nonconvex (QP1QC) problem can be solved through a dual approach with no duality gap. This is unusual for general nonconvex programming but we can explain by showing that (QP1QC) is indeed equivalent to a linearly constrained convex problem, which happens to be dual of the dual of itself. Another type of hidden convexity can be also found in the boundarification technique developed in Xing et?al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted).     

Finding all solutions of separable systems of piecewise-linear equations using integer programming

Finding all solutions of nonlinear or piecewise-linear equations is an important problem which is widely encountered in science and engineering. Various algorithms have been proposed for this problem. However, the implementation of these algorithms are generally difficult for non-experts or beginners. In this paper, an efficient method is proposed for finding all solutions of separable systems of piecewise-linear equations using integer programming. In this method, we formulate the problem of finding all solutions by a mixed integer programming problem, and solve it by a high-performance integer programming software such as GLPK, SCIP, or CPLEX. It is shown that the proposed method can be easily implemented without making complicated programs. It is also confirmed by numerical examples that the proposed method can find all solutions of medium-scale systems of piecewise-linear equations in practical computation time.     

Neuro-fuzzy based constraint programming

Constraint programming models appear in many sciences including mathematics, engineering and physics. These problems aim at optimizing a cost function joint with some constraints. Fuzzy constraint programming has been developed for treating uncertainty in the setting of optimization problems with vague constraints. In this paper, a new method is presented into creation fuzzy concept for set of constraints. Unlike to existing methods, instead of constraints with fuzzy inequalities or fuzzy coefficients or fuzzy numbers, vague nature of constraints set is modeled using learning scheme with adaptive neural-fuzzy inference system (ANFIS). In the proposed approach, constraints are not limited to differentiability, continuity, linearity; also the importance degree of each constraint can be easily applied. Unsatisfaction of each weighted constraint reduces membership of certainty for set of constraints. Monte-Carlo simulations are used for generating feature vector samples and outputs for construction of necessary data for ANFIS. The experimental results show the ability of the proposed approach for modeling constrains and solving parametric programming problems.     

Nonsmooth multiobjective programming and constraint qualifications

We consider a nonsmooth multiobjective programming problem with inequality and set constraints. By using the notion of convexificator, we extend the Abadie constraint qualification, and derive the strong Kuhn-Tucker necessary optimality conditions. Some other constraint qualifications have been generalized and their interrelations are investigated.     

Integrating operations research in constraint programming

This paper presents constraint programming (CP) as a natural formalism for modelling problems, and as a flexible platform for solving them. CP has a range of techniques for handling constraints including several forms of propagation and tailored algorithms for global constraints. It also allows linear programming to be combined with propagation and novel and varied search techniques which can be easily expressed in CP. The paper describes how CP can be used to exploit linear programming within different kinds of hybrid algorithm. In particular it can enhance techniques such as Lagrangian relaxation, Benders decomposition and column generation.     

A constraint qualification for convex programming

1. IntroductionLet X be the n-dimensional Euclidean space Rad. Denote the transpose of the columnvector y by yT. Suppose that R7 and nit R= are the n-dimensional vector sets with nonnegative and positive components, whose elements are denoted by y 3 0 and y > 0, respectively.Write R for the nonnegative real number set.Consider a nondiffereatiable convex programming problem:We assume that f(x), gi(x),..', g.(x) are finite reaLvalued continuous, convex functions on X, but not necessarily d…     

On constraint qualifications in nonlinear programming

In this paper we give conditions for deriving the inconsistency of an inequality system of positively homogeneous functions starting from the inconsistency of another one. When the impossibility of the starting system represents a necessary optimality condition for an inequality constrained extremum problem and the positively homogeneous functions involved have suitable properties of convexity, such conditions collapse into the well known constraint qualifications.     

General constraint qualifications in nondifferentiable programming

We show that a familiar constraint qualification of differentiable programming has nonsmooth counterparts. As a result, necessary optimality conditions of Kuhn—Tucker type can be established for inequality-constrained mathematical programs involving functions not assumed to be differentiable, convex, or locally Lipschitzian. These optimality conditions reduce to the usual Karush—Kuhn—Tucker conditions in the differentiable case and sharpen previous results in the locally Lipschitzian case.     

A probabilistic programming model for blending aggregates

The paper introduces the formulation of a probabilistic programming model to find the optimum mix proportion of aggregates to meet the specific grading requirement in order to minimize the cost which consists of the material cost and the expected penalty cost. The model is probabilistic since the gradation, which is the major parameter, is a random variable. A linear programming model is first formulated. Using the LP solution as initial value, a direct search technique is then employed to solve the problem. The model is expected to be applicable to any problem of aggregates blending. In this paper, however, the mixing aggregates of an asphalt mixing plant is exemplified to test the applicability of the model.     

Uniform quasi-concavity in probabilistic constrained stochastic programming

A probabilistic constrained stochastic linear programming problem is considered, where the rows of the random technology matrix are independent and normally distributed. The quasi-concavity of the constraining function needed for the convexity of the problem is ensured if the factors of the function are uniformly quasi-concave. A necessary and sufficient condition is given for that property to hold. It is also shown, through numerical examples, that such a special problem still has practical application in optimal portfolio construction.     

Combining metaheuristics with mathematical programming, constraint programming and machine learning

During the last years, interest on hybrid metaheuristics has risen considerably in the field of optimization and machine learning. The best results found for many optimization problems in science and industry are obtained by hybrid optimization algorithms. Combinations of optimization tools such as metaheuristics, mathematical programming, constraint programming and machine learning, have provided very efficient optimization algorithms. Four different types of combinations are considered in this paper: (i) Combining metaheuristics with complementary metaheuristics. (ii) Combining metaheuristics with exact methods from mathematical programming approaches which are mostly used in the operations research community. (iii) Combining metaheuristics with constraint programming approaches developed in the artificial intelligence community. (iv) Combining metaheuristics with machine learning and data mining techniques.     

Combining metaheuristics with mathematical programming,constraint programming and machine learning

We consider the problem of creating fair course timetables in the setting of a university. The central idea is that undesirable arrangements in the course timetable, i.e., violations of soft constraints, should be distributed in a fair way among the stakeholders. We propose and discuss in detail two fair versions of the popular curriculum-based course timetabling (CB-CTT) problem, the MMF-CB-CTT problem and the JFI-CB-CTT problem, which are based on max–min fairness (MMF) and Jain’s fairness index (JFI), respectively. For solving the MMF-CB-CTT problem, we present and experimentally evaluate an optimization algorithm based on simulated annealing. We introduce three different energy difference measures and evaluate their impact on the overall algorithm performance. The proposed algorithm improves the fairness on 20 out of 32 standard instances compared to the known best timetables. The JFI-CB-CTT problem formulation focuses on the trade-off between fairness and the aggregated soft constraint violations. Here, our experimental evaluation shows that the known best solutions to 32 CB-CTT standard instances are quite fair with respect to JFI. Our experiments show that the fairness can often be improved at the cost of only a small increase in the overall amount of penalty.     

A simple constraint qualification in convex programming

We introduce and characterize a class of differentiable convex functions for which the Karush—Kuhn—Tucker condition is necessary for optimality. If some constraints do not belong to this class, then the characterization of optimality generally assumes an asymptotic form.We also show that for the functions that belong to this class in multi-objective optimization, Pareto solutions coincide with strong Pareto solutions,. This extends a result, well known for the linear case.Research partly supported by the National Research Council of Canada.     

Adaptive constraint reduction for convex quadratic programming

We propose an adaptive, constraint-reduced, primal-dual interior-point algorithm for convex quadratic programming with many more inequality constraints than variables. We reduce the computational effort by assembling, instead of the exact normal-equation matrix, an approximate matrix from a well chosen index set which includes indices of constraints that seem to be most critical. Starting with a large portion of the constraints, our proposed scheme excludes more unnecessary constraints at later iterations. We provide proofs for the global convergence and the quadratic local convergence rate of an affine-scaling variant. Numerical experiments on random problems, on a data-fitting problem, and on a problem in array pattern synthesis show the effectiveness of the constraint reduction in decreasing the time per iteration without significantly affecting the number of iterations. We note that a similar constraint-reduction approach can be applied to algorithms of Mehrotra’s predictor-corrector type, although no convergence theory is supplied.     

Fuzzy number linear programming: A probabilistic approach (3)

In the real world there are many linear programming problems where all decision parameters are fuzzy numbers. Several approaches exist which use different ranking functions for solving these problems. Unfortunately when there exist alternative optimal solutions, usually with different fuzzy value of the objective function for these solutions, these methods can not specify a clear approach for choosing a solution. In this paper we propose a method to remove the above shortcoming in solving fuzzy number linear programming problems using the concept of expectation and variance as ranking functions.     

Finding the projection of a given point on the set of solutions of a linear programming problem

The problem of finding the projections of points on the sets of solutions of primal and dual problems of linear programming is considered. This problem is reduced to a single solution of the problem of minimizing a new auxiliary function, starting from some threshold value of the penalty coefficient. Estimates of the threshold value are obtained. A software implementation of the proposed method is compared with some known commercial and research software packages for solving linear programming problems.     

Integrated production and material handling scheduling using mathematical programming and constraint programming

In this article, we propose an integrated formulation of the combined production and material handling scheduling problems. Traditionally, scheduling problems consider the production machines as the only constraining resource. This is however no longer true as material handling vehicles are becoming more and more valuable resources requiring important investments. Their operations should be optimized and above all synchronized with machine operations. In the problem addressed in this paper, a job shop context is considered. Machines and vehicles are both considered as constraining resources. The integrated scheduling problem is formulated as a mathematical programming model and as a constraint programming model which are compared for optimally solving a series of test problems. A commercial software (ILOG OPLStudio) was used for modeling and testing both models.     

Enhancing the performance of constraint programming through the introduction of linear programming

Constraint Programming (CP) has been successful in a number of combinatorial search and discrete optimisation problems. Yet other more traditional approaches, such as Integer Programming (IP), can still give a better performance on the same problem types. Central to IP's success is its reliance on a fast Linear Programming (LP) solver providing solutions during the search to the corresponding relaxed problems. These solutions are used to guide the search within IP as well as a means of detecting infeasibility and integrality. This paper shows that there is scope also to include LP within the CP framework, in order to similarly guide the CP search. The problems examined here are one for which CP on its own had proved markedly inferior to IP. Hence a hybrid solver based on the CP search and using an LP solver is configured and run on these problems. The outcome shows that using the LP solver within the CP search is a valuable addition to the available search strategies. An improved performance over the CP-only strategies is obtained and, further, comparable results are obtained to those from IP. Overall, CP+LP can be considered as a more robust approach than either CP or IP on their own on a variety of combinatorial search problems.