An interactive method for multiple-objective mathematical programming problems

An interactive method is developed for solving the general nonlinear multiple objective mathematical programming problems. The method asks the decision maker to provide partial information (local tradeoff ratios) about his utility (preference) function at each iteration. Using the information, the method generates an efficient solution and presents it to the decision maker. In so doing, the best compromise solution is sought in a finite number of iterations. This method differs from the existing feasible direction methods in that (i) it allows the decision maker to consider only efficient solutions throughout, (ii) the requirement of line search is optional, and (iii) it solves the problems with linear objective functions and linear utility function in one iteration. Using various problems selected from the literature, five line search variations of the method are tested and compared to one another. The nonexisting decision maker is simulated using three different recognition levels, and their impact on the method is also investigated.     

Strange: An interactive method for multi-objective linear programming under uncertainty

In the field of investment planning within a time horizon, problems typically involve multiple objectives, and basic data are uncertain. In a large number of cases, these decision problems can be written as linear programming problems in which time dependent uncertainties affect the coefficients and the right hand side of constraints. Given the possibility of defining plausible scenarios on basic data, discrete sets of such coefficients are given, each with its subjective probability of occurrence. The corresponding structure is then characteristic for Multi-Objective Stochastic Linear Programming (MOSLP).In the paper, an interactive procedure is described to obtain a best compromise for such a MOSLP problem. This algorithm, called Strange, extends the Stem method to take the random aspects into account. It involves in particular, the concepts of stochastic programming with recourse. In its interactive steps, the efficiency projection techniques are used to provide the decision-maker with detailed graphical information on efficient solution families.As an illustration of the successive steps, a didactic example is solved in some detail, and the results of a case study in energy planning are given.     

An efficient method for solving linear goal programming problems

This note proposes a solution algorithm for linear goal programming problems. The proposed method simplifies the traditional solution methods. Also, the proposed method is computationally efficient.     

A new method of solving nonlinear mathematical programming problems involving r-invex functions

A new approach to a solution of a nonlinear constrained mathematical programming problem involving r-invex functions with respect to the same function η is introduced. An η-approximated problem associated with an original nonlinear mathematical programming problem is presented that involves η-approximated functions constituting the original problem. The equivalence between optima points for the original mathematical programming problem and its η-approximated optimization problem is established under r-invexity assumption.     

Numerical algorithm for solving mathematical programming problems with a smooth surface as a constraint

A numerical algorithm for minimizing a convex function on a smooth surface is proposed. The algorithm is based on reducing the original problem to a sequence of convex programming problems. Necessary extremum conditions are examined, and the convergence of the algorithm is analyzed.     

An interactive fuzzy satisficing method for multiobjective stochastic linear programming problems through an expectation model

For decision making problems involving uncertainty, both stochastic programming as an optimization method based on the theory of probability and fuzzy programming representing the ambiguity by fuzzy concept have been developing in various ways. In this paper, we focus on multiobjective linear programming problems with random variable coefficients in objective functions and/or constraints. For such problems, as a fusion of these two approaches, after incorporating fuzzy goals of the decision maker for the objective functions, we propose an interactive fuzzy satisficing method for the expectation model to derive a satisficing solution for the decision maker. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

An effective method for solving linear programming problems with flexible constraints

For solving linear programming problems with flexible constraints being specific piecewise linear programs a new algorithm which is based on a systematic decomposition of the feasible set into linear constrained subsets is proposed and shown to converge.     

An extraproximal method for solving equilibrium programming problems in a Hilbert space

A regularized method of the proximal type for solving equilibrium problems in a Hilbert space is proposed. The method is combined with an approximation of the original problem. The convergence of the method is analyzed, and a regularizing operator is constructed.     

Shooting method for solving equilibrium programming problems

A new iterative method is proposed for solving equilibrium programming problems. The sequence of points it generates is proved to converge weakly to the solution set of the equilibrium problem under study. If the initial point has at least one projection onto the solution set of the equilibrium problem, the sequence generated by the method is shown to converge strongly to the set of these projections. The partial gradient of the initial data is assumed to be invertible and strictly monotone, which differs from the classical skew-symmetry condition.     

An approximation algorithm for convex multi-objective programming problems

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson’s outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

A hybrid method for solving multi-objective global optimization problems

Real optimization problems often involve not one, but multiple objectives, usually in conflict. In single-objective optimization there exists a global optimum, while in the multi-objective case no optimal solution is clearly defined but rather a set of optimums, which constitute the so called Pareto-optimal front. Thus, the goal of multi-objective strategies is to generate a set of non-dominated solutions as an approximation to this front. However, most problems of this kind cannot be solved exactly because they have very large and highly complex search spaces. The objective of this work is to compare the performance of a new hybrid method here proposed, with several well-known multi-objective evolutionary algorithms (MOEA). The main attraction of these methods is the integration of selection and diversity maintenance. Since it is very difficult to describe exactly what a good approximation is in terms of a number of criteria, the performance is quantified with adequate metrics that evaluate the proximity to the global Pareto-front. In addition, this work is also one of the few empirical studies that solves three-objective optimization problems using the concept of global Pareto-optimality.     

A review of Hopfield neural networks for solving mathematical programming problems

The Hopfield neural network (HNN) is one major neural network (NN) for solving optimization or mathematical programming (MP) problems. The major advantage of HNN is in its structure can be realized on an electronic circuit, possibly on a VLSI (very large-scale integration) circuit, for an on-line solver with a parallel-distributed process. The structure of HNN utilizes three common methods, penalty functions, Lagrange multipliers, and primal and dual methods to construct an energy function. When the function reaches a steady state, an approximate solution of the problem is obtained. Under the classes of these methods, we further organize HNNs by three types of MP problems: linear, non-linear, and mixed-integer. The essentials of each method are also discussed in details. Some remarks for utilizing HNN and difficulties are then addressed for the benefit of successive investigations. Finally, conclusions are drawn and directions for future study are provided.     

The basis for an operational method of solving certain problems in mathematical physics

In this paper we obtain sufficient conditions for the applicability of an operational method for solving a mixed problem for an equation of parabolic type with discontinuous coefficients.Translated from Matematicheskie Zametki, Vol. 13, No. 1, pp. 125–134, January, 1973.     

An unconstrained convex programming approach to solving convex quadratic programming problems

In this paper, we derive an unconstrained convex programming approach to solving convex quadratic programming problems in standard form. Related duality theory is established by using two simple inequalities. An ?-optimal solution is obtained by solving an unconstrained dual convex program. A dual-to-primal conversion formula is also provided. Some preliminary computational results of using a curved search method is included     

An outranking-based general approach to solving group multi-objective optimization problems

This paper presents a general approach to solving multi-objective programming problems with multiple decision makers. The proposal is based on optimizing a bi-objective measure of “collective satisfaction”. Group satisfaction is understood as a reasonable balance between the strengths of an agreeing and an opposing coalition, considering also the number of decision makers not belonging to any of these coalitions. Accepting the vagueness of “collective satisfaction”, even the vagueness of “person satisfaction”, fuzzy outranking relations and other fuzzy logic models are used.     

Comparative studies in interactive multiple objective mathematical programming

In the past two decades, there has been a significant increase in the number of interactive algorithms proposed for solving multiple objective mathematical programming (MOMP) problems. Most of these procedures have neither been tested in real decision making situations, nor compared to each other. In this study, we emphasize the importance of comparative studies of interactive MOMP procedures and present a state of the art review. Our scope is limited to the comparisons of interactive procedures for solving deterministic, linear, integer or nonlinear constrained multiple objective optimization problems involving a single decision maker.     

On the method of multipliers for mathematical programming problems

In this paper, the numerical solution of the basic problem of mathematical programming is considered. This is the problem of minimizing a functionf(x) subject to a constraint (x)=0. Here,f is a scalar,x is ann-vector, and is aq-vector, withq<n.The approach employed is based on the introduction of the augmented penalty functionW(x,,k)=f(x)+ ^T (x)+k ^T (x) (x). Here, theq-vector is an approximation to the Lagrange multiplier, and the scalark>0 is the penalty constant.Previously, the augmented penalty functionW(x, ,k) was used by Hestenes in his method of multipliers. In Hestenes' version, the method of multipliers involves cycles, in each of which the multiplier and the penalty constant are held constant. After the minimum of the augmented penalty function is achieved in any given cycle, the multiplier is updated, while the penalty constantk is held unchanged.In this paper, two modifications of the method of multipliers are presented in order to improve its convergence characteristics. The improved convergence is achieved by (i) increasing the updating frequency so that the number of iterations in a cycle is shortened to N=1 for the ordinary-gradient algorithm and the modified-quasilinearization algorithm and N=n for the conjugate-gradient algorithm, (ii) imbedding Hestenes' updating rule for the multiplier into a one-parameter family and determining the scalar parameter so that the error in the optimum condition is minimized, and (iii) updating the penalty constantk so as to cause some desirable effect in the ordinary-gradient algorithm, the conjugate-gradient algorithm, and the modified-quasilinearization algorithm. For the sake of identification, Hestenes' method of multipliers is called Method MM-1, the modification including (i) and (ii) is called Method MM-2, and the modification including (i), (ii), (iii) is called Method MM-3.Evaluation of the theory is accomplished with seven numerical examples. The first example pertains to a quadratic function subject to linear constraints. The remaining examples pertain to non-quadratic functions subject to nonlinear constraints. Each example is solved with the ordinary-gradient algorithm, the conjugate-gradient algorithm, and the modified-quasilinearization algorithm, which are employed in conjunction with Methods MM-1, MM-2, and MM-3.The numerical results show that (a) for given penalty constantk, Method MM-2 generally exhibits faster convergence than Method MM-1, (b) in both Methods MM-1 and MM-2, the number of iterations for convergence has a minimum with respect tok, and (c) the number of iterations for convergence of Method MM-3 is close to the minimum with respect tok of the number of iterations for convergence of Method MM-2. In this light, Method MM-3 has very desirable characteristics.This research was supported by the National Science Foundation, Grant No. GP-32453. The authors are indebted to Messieurs E. E. Cragg and A. Esterle for computational assistance.     

An improved sequential quadratic programming algorithm for solving general nonlinear programming problems

In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising.     

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