On accuracy,robustness and tolerances in vector Boolean optimization

A Boolean programming problem with a finite number of alternatives where initial coefficients (costs) of linear payoff functions are subject to perturbations is considered. We define robust solution as a feasible solution which for a given set of realizations of uncertain parameters guarantees the minimum value of the worst-case relative regret among all feasible solutions. For the Pareto optimality principle, an appropriate definition of the worst-case relative regret is specified. It is shown that this definition is closely related to the concept of accuracy function being recently intensively studied in the literature. We also present the concept of robustness tolerances of a single cost vector. The tolerance is defined as the maximum level of perturbation of the cost vector which does not destroy the solution robustness. We present formulae allowing the calculation of the robustness tolerance obtained for some initial costs. The results are illustrated with several numerical examples.     

Global Optimization of Nonlinear Bilevel Programming Problems

A novel technique that addresses the solution of the general nonlinear bilevel programming problem to global optimality is presented. Global optimality is guaranteed for problems that involve twice differentiable nonlinear functions as long as the linear independence constraint qualification condition holds for the inner problem constraints. The approach is based on the relaxation of the feasible region by convex underestimation, embedded in a branch and bound framework utilizing the basic principles of the deterministic global optimization algorithm, BB [2, 4, 5, 11]. Epsilon global optimality in a finite number of iterations is theoretically guaranteed. Computational studies on several literature problems are reported.     

A weak linearization of the Kuhn-tucker conditions for a class of allocation problems

The return obtained from the allocation of resources to an activity is occasionally modelled by means of concave, strictly increasing functions. Exponential functions of a certain class conveniently lend themselves to such modelling. A nonlinear programming formulation of a multiresource allocation problem with return functions of the class appears to have Kuhn-Tucker conditions which in a sense are intrinsically linear. The paper shows how this fact can be utilised to save processing time in the execution of numerical algorithms for the solution of this mathematical programming problem.     

Solving convex quadratic bilevel programming problems using an enumeration sequential quadratic programming algorithm

In this paper, we present an original method to solve convex bilevel programming problems in an optimistic approach. Both upper and lower level objective functions are convex and the feasible region is a polyhedron. The enumeration sequential linear programming algorithm uses primal and dual monotonicity properties of the primal and dual lower level objective functions and constraints within an enumeration frame work. New optimality conditions are given, expressed in terms of tightness of the constraints of lower level problem. These optimality conditions are used at each step of our algorithm to compute an improving rational solution within some indexes of lower level primal-dual variables and monotonicity networks as well. Some preliminary computational results are reported.     

Descent methods for quasidifferentiable minimization

A descent method is given for minimizing a nondifferentiable function which can be locally approximated by pointwise minima of convex functions. At each iterate the algorithm finds several directions by solving several linear or quadratic programming subproblems. These directions are then used in an Armijo-like search for the next iterate. A feasible direction extension to inequality constrained minimization problems is also presented. The algorithms converge to points satisfying necessary optimality conditions which are sharper than the ones involved in convergence results for algorithms based on the Clarke subdifferential.This research was sponsored by Project 02.15.     

Geometric and algorithmic developments for a hierarchical planning problem

This paper presents a new model for multiobjective planning in hierarchical systems that explicitly takes into consideration the order in which decisions are made. Interactions and conflicts that normally exist among the levels are introduced by specifying jointly controlled feasible regions and interdependent objective functions. At each level in the system, planners attempt to maximize net benefits in light of all higher-level decisions, and thus may influence but not control the behavior of others. The resultant formulation leads to the multilevel programming problem. The geometry of an all linear case is first examined wherein it is shown that the optimal solution must lie at a vertex of the original polyhedral constraint region. Next, a set of first order optimality conditions is derived for the general case and used as the basis of an algorithm for the linear problem. A number of examples are given to highlight the results.     

Sufficient Optimality Criterion for Linearly Constrained,Separable Concave Minimization Problems

A sufficient optimality criterion for linearly-constrained concave minimization problems is given in this paper. Our optimality criterion is based on the sensitivity analysis of the relaxed linear programming problem. The main result is similar to that of Phillips and Rosen (Ref. 1); however, our proofs are simpler and constructive.In the Phillips and Rosen paper (Ref. 1), they derived a sufficient optimality criterion for a slightly different linearly-constrained concave minimization problem using exponentially many linear programming problems. We introduce special test points and, using these for several cases, we are able to show optimality of the current basic solution.The sufficient optimality criterion described in this paper can be used as a stopping criterion for branch-and-bound algorithms developed for linearly-constrained concave minimization problems.This research was supported by a Bolyai János Research Fellowship BO/00334/00 of the Hungarian Academy of Science and by the Hungarian Scientific Research Foundation, Grant OTKA T038027.     

Bounded knapsack sharing

A bounded knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more linear inequality constraints, an objective function composed of single variable continuous functions called tradeoff functions, and lower and upper bounds on the variables. A single constraint problem which can have negative or positive constraint coefficients and any type of continuous tradeoff functions (including multi-modal, multiple-valued and staircase functions) is considered first. Limiting conditions where the optimal value of a variable may be plus or minus infinity are explicitly considered. A preprocessor procedure to transform any single constraint problem to a finite form problem (an optimal feasible solution exists with finite variable values) is developed. Optimality conditions and three algorithms are then developed for the finite form problem. For piecewise linear tradeoff functions, the preprocessor and algorithms are polynomially bounded. The preprocessor is then modified to handle bounded knapsack sharing problems with multiple constraints. An optimality condition and algorithm is developed for the multiple constraint finite form problem. For multiple constraints, the time needed for the multiple constraint finite form algorithm is the time needed to solve a single constraint finite form problem multiplied by the number of constraints. Some multiple constraint problems cannot be transformed to multiple constraint finite form problems.     

Incremental allocation of a discrete and convex resource to concave activities

The problem considered is that of the allocation of a given quantity of a discrete resource to activities described by concave return functions. The resource-consumption corresponding to each allocation is described by a convex function of the quantity allocated. An incremental solution algorithm is given, which specializes to the algorithm of Fox if the resource is linear, and to an algorithm of Katoh, Ibaraki and Mine if the return functions are linear.     

Discrete allocation of a linear resource according to a sum-of-products objective

The allocation of a linear resource according to the sum of the returns from independent activities is considered. The return from each activity is given by a product of concave and nondecreasing functions of a single allocation variable. The model can be used, for instance, to describe probabilities of success of several serial tasks, into which an activity is subdivided. An incremental algorithm is defined and conditions are given for the algorithm to generate an optimal solution; otherwise, the problem is solved by a two-step procedure involving the incremental maximization of the return corresponding to a single activity and the combination of the activities by dynamic programming. Examples are given of problems solvable and not solvable by the incremental algorithm.     

Vertex packings: Structural properties and algorithms

We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.     

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.     

Optimality and duality for multiple-objective optimization under generalized type I univexity

In this paper, we extend the classes of generalized type I vector-valued functions introduced by Aghezzaf and Hachimi in [J. Global Optim. 18 (2000) 91-101] to generalized univex type I vector-valued functions and consider a multiple-objective optimization problem involving generalized type I univex functions. A number of Kuhn-Tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution. The Mond-Weir and general Mond-Weir type duality results are also presented.     

Coupled problems of the optimization of the elasto-plastic deformation of metals

The mathematical formulation of a coupled optimization problem, which includes several optimal problems which are interrelated via the optimization parameters and functions of state, is analysed. The conditions for the existence of a solution are formulated and a technique for reducing the problem, to a sequence of optimal problems with unknown functions in the optimality and bounding criteria is described. A special minimizing sequence is constructed and its convergence conditions are derived. An example is given to demonstrate the method.     

线性多级规划的最优性条件和基本性质

本文研究的线性多级规划模型比较一般化，容许集可以是无界的，每级的目标函数可以与各下级控制的决策变量有关．我们得到了这类多级规划的一组最优性充要条件，利用这组条件推导了各级可行集的弱拟凸性、连通性等几何性质．作为应用订正了Bard的一个例题．     

Discrete Allocation and Replenishment of Resources with Binary Effectiveness

The problem considered is that of the allocation and replenishment of several resources in integer quantities in such a way as to maximize the sum of the returns from activities with concave return functions. All the resources are of the same physical type and each resource has an effectiveness of 0 or 1 against each activity, depending on the geographical locations of the resources and the activities or on other constraints. Solutions with objective values arbitrarily close to the optimal value are generated by the application of resourcewise optimization to an associated problem in continuous variables, and the rounding of a continuous solution to an integer solution according to given rules. The application of other continuous methods is indicated. Some properties of optimal integer solutions are derived.     

Allocation of dependent divisional resources

A resource allocation problem is considered with resources that are dependent in the sense that an allocation to an activity requires the application of several resources, except for certain activities which are divisional in the sense that an allocation to such an activity requires the use of only a single resource. Return and cost functions are assumed to be continuous and increasing, and the allocation variables are continuous. Conditions are given for the replacement of the continuous problem by an associated problem with discrete variables and a single constraint, and to a given degree of accuracy. The associated problem can be efficiently solved by dynamic programming. Certain divisional resource allocation problems with discrete variables and several linear constraints are shown to be equivalent to a discrete problem with a single constraint. A numerical example is given.     

Some nondifferentiable multiobjective programming under generalized d-V-type-I univexity

In this paper, we introduce new classes of functions called d-V-type-I univex by extending the definition of d-V-type-I functions and consider a multiobjective optimization problem involving generalized d-V-type-I univex functions. A number of Karush–Kuhn–Tucker-type sufficient optimality conditions are obtained for a feasible solution to be a weak Pareto efficient solution. The Mond–Weir-type duality results are also presented. The results obtained in this paper generalize and extend the previously known result in this area.     

Improving benders decomposition using a genetic algorithm

We develop and investigate the performance of a hybrid solution framework for solving mixed-integer linear programming problems. Benders decomposition and a genetic algorithm are combined to develop a framework to compute feasible solutions. We decompose the problem into a master problem and a subproblem. A genetic algorithm along with a heuristic are used to obtain feasible solutions to the master problem, whereas the subproblem is solved to optimality using a linear programming solver. Over successive iterations the master problem is refined by adding cutting planes that are implied by the subproblem. We compare the performance of the approach against a standard Benders decomposition approach as well as against a stand-alone solver (Cplex) on MIPLIB test problems.     

Evaluation and Incremental Determination of Almost Optimal Allocations of Resources

The problem of allocating resources to activities with strictly concave return functions is considered; the objective function to be maximized is the sum of the returns from each activity. It is demonstrated that any set of feasible allocations can be used to obtain an explicit upper bound of the optimal value of this function. The upper bound is used to check that a numerically fast incremental procedure produces almost optimal allocations. A conservative solution of the allocation problem is generated by successively incrementing allocations with the greatest marginal returns; practical allocations are obtained from the conservative allocations by a method resulting in a reduction of the number of nonzero allocations and a simultaneous increase of the value of the objective function.