On the complexity of inverse convex ordered 1-median problem on the plane and on tree networks

An ordered median function is used in location theory to generalize a class of problems, including median and center problems. In this paper we consider the complexity of inverse ordered 1-median problems on the plane and on trees, where the multipliers are sorted nondecreasingly. Based on the convexity of the objective function, we prove that the problems with variable weights or variable coordinates on the line are NP-hard. Then we can directly get the NP-hardness result for the corresponding problem on the plane. We finally develop a cubic time algorithm that solves the inverse convex ordered 1-median problem on trees with relaxation on modification bounds.     

Stochastic analysis of ordered median problems

Many location problems can be expressed as ordered median objective. In this paper, we investigate the ordered median objective when the demand points are generated in a circle. We find the mean and variance of the kth distance from the centre of the circle and the correlation matrix between all pairs of ordered distances. By applying these values, we calculate the mean and variance of any ordered median objective and the correlation coefficient between two ordered median objectives. The usefulness of the results is demonstrated by calculating various probabilities such as: What is the probability that the mean distance is greater than the truncated mean distance? What is the probability that the maximum distance is greater than 0.9? What is the probability that the range of distances is greater than 0.8? An analysis of an illustrative example also demonstrates the usefulness of the analysis.     

An Approximation Scheme for Uncertain Minimax Optimal Control Problems

In this work, we address an uncertain minimax optimal control problem with linear dynamics where the objective functional is the expected value of the supremum of the running cost over a time interval. By taking an independently drawn random sample, the expected value function is approximated by the corresponding sample average function. We study the epi-convergence of the approximated objective functionals as well as the convergence of their global minimizers. Then we define an Euler discretization in time of the sample average problem and prove that the value of the discrete time problem converges to the value of the sample average approximation. In addition, we show that there exists a sequence of discrete problems such that the accumulation points of their minimizers are optimal solutions of the original problem. Finally, we propose a convergent descent method to solve the discrete time problem, and show some preliminary numerical results for two simple examples.     

The -digraph optimization problem and the greedy algorithm

In this paper we present a new optimization problem and a general class of objective functions for this problem. We show that optimal solutions to this problem with these objective functions are found with a simple greedy algorithm. Special cases include matroids, Huffman's data compression problem, a special class of greedoids, a special class of min cost max flow problems (related to Monge sequences), a special class of weighted f-factor problems, and some new problems.     

A new class of alternative theorems for SOS-convex inequalities and robust optimization

In this paper, we present a new class of alternative theorems for SOS-convex inequality systems without any qualifications. This class of theorems provides an alternative equations in terms of sums of squares to the solvability of the given inequality system. A strong separation theorem for convex sets, described by convex polynomial inequalities, plays a key role in establishing the class of alternative theorems. Consequently, we show that the optimal values of various classes of robust convex optimization problems are equal to the optimal values of related semidefinite programming problems (SDPs) and so, the value of the robust problem can be found by solving a single SDP. The class of problems includes programs with SOS-convex polynomials under data uncertainty in the objective function such as uncertain quadratically constrained quadratic programs. The SOS-convexity is a computationally tractable relaxation of convexity for a real polynomial. We also provide an application of our theorem of the alternative to a multi-objective convex optimization under data uncertainty.     

Production of locomotive rosters for a multi-class multi-locomotive problem

This paper describes a problem faced by the Public Transport Corporation (PTC) in the Australian State of Victoria. It involved the allocation of locomotives to the freight trains operated by the Corporation. Such a problem can be classified as a multi-class multi-locomotive problem since different types (or classes) of locomotive are available to pull the trains and some of the trains are heavy enough to require more than one locomotive. Because of the features of the services operated by the PTC, formulation of this problem as a pure integer program is straightforward, provided the objective function can be linearised. However, obtaining the optimal solution from this formulation is not. Large optimality gaps remained after various attempts to tighten the problem. Other authors have reported similar difficulties with similar problems. The original problem was therefore reformulated using special ordered covering sets of type 1. After reformulation, the optimal solution was obtained in less than a second of CPU time. This enabled alternative methods of linearising the objective function to be evaluated, resulting eventually in the decision to extend the concept of minimal covering sets so that each integer variable in the problem was expressed as a linear sum of a special ordered covering set of binary variables. In this way, the objective function was both exact and linear and the model, although much larger, still solved in under a second of CPU time. The methods of solution proposed in this paper should be capable of extension to more general resource allocation problems exhibiting similar features.     

A revised bound improvement sequence algorithm

In this paper we present a generalization and a computational improvement of the Bound Improvement Sequence Algorithm. The main computational burden of this algorithm consists in determining whether there exists a feasible point on the objective hyperplane, when the algorithm encounters a fixed point. By generalizing the algorithm, such that the objective function and constraints are treated alike, the number of fixed points that are required can be reduced. The computational results that we report allow us to conclude that the number of fixed points can generally be reduced for loosely constrained problems. For this class of problems the new algorithm appears to be more efficient than a standard MIP code such as FMPS.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

An algebraic approach to assignment problems

For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.     

Thep-facility ordered median problem on networks

In this paper we deal with the ordered median problem: a family of location problems that allows us to deal with a large number of real situations which does not fit into the standard models of location analysis. Moreover, this family includes as particular instances many of the classical location models. Here, we analyze thep-facility version of this problem on networks and our goal is to study the structure of the set of candidate points to be optimal solutions. The research of the authors is partially financed by Spanish research grants BFM2001-2378, BFM2001-4028, BFM2004-0909 and HA2003-0121.     

基于重新排序的退化工件最小化总延误时间问题

考虑了错位限制下的含有退化工件的重新排序问题,即工件的实际加工时间看作是工件开工时间的线性函数．重新排序就是在原始工件已经按照某种规则使目标函数达到最优时有一新工件集到达,新工件的安排使得原始工件重新排序进而产生错位．研究了最大序列错位和总序列错位限制下的退化工件最小化总延误时间问题,其最优排序的结构性质是使得原始工件集和新工件集中的工件是按加工率αj非减的序列排列,基于此通过分阶段排序和动态规划方法给出了两个问题的多项式时间的最优算法．     

A path following method for box-constrained multiobjective optimization with applications to goal programming problems

We propose a path following method to find the Pareto optimal solutions of a box-constrained multiobjective optimization problem. Under the assumption that the objective functions are Lipschitz continuously differentiable we prove some necessary conditions for Pareto optimal points and we give a necessary condition for the existence of a feasible point that minimizes all given objective functions at once. We develop a method that looks for the Pareto optimal points as limit points of the trajectories solutions of suitable initial value problems for a system of ordinary differential equations. These trajectories belong to the feasible region and their computation is well suited for a parallel implementation. Moreover the method does not use any scalarization of the multiobjective optimization problem and does not require any ordering information for the components of the vector objective function. We show a numerical experience on some test problems and we apply the method to solve a goal programming problem.     

Enumeration of All Possibly Optimal Vertices with Possible Optimality Degrees in Linear Programming Problems with a Possibilistic Objective Function

In this paper, we treat linear programming problems with fuzzy objective function coefficients. To such a problem, the possibly optimal solution set is defined as a fuzzy set. It is shown that any possibly optimal solution can be represented by a convex combination of possibly optimal vertices. A method to enumerate all possibly optimal vertices with their membership degrees is developed. It is shown that, given a possibly optimal extreme point with a higher membership degree, the membership degree of an adjacent extreme point is calculated by solving a linear programming problem and that all possibly optimal vertices are enumerated sequentially by tracing adjacent possibly optimal extreme points from a possibly optimal extreme point with the highest membership degree.     

A filled function method for optimal discrete-valued control problems

In this paper, we present a new approach to solve a class of optimal discrete-valued control problems. This type of problem is first transformed into an equivalent two-level optimization problem involving a combination of a discrete optimization problem and a standard optimal control problem. The standard optimal control problem can be solved by existing optimal control software packages such as MISER 3.2. For the discrete optimization problem, a discrete filled function method is developed to solve it. A numerical example is solved to illustrate the efficiency of our method.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Ordered median problem with demand distribution weights

The ordered median function unifies and generalizes most common objective functions used in location theory. It is based on the ordered weighted averaging (OWA) operator with the preference weights allocated to the ordered distances. Demand weights are used in location problems to express the client demand for a service thus defining the location decision output as distances distributed according to measures defined by the demand weights. Typical ordered median model allows weighting of several clients only by straightforward rescaling of the distance values. However, the OWA aggregation of distances enables us to introduce demand weights by rescaling accordingly clients measure within the distribution of distances. It is equivalent to the so-called weighted OWA (WOWA) aggregation of distances covering as special cases both the weighted median solution concept defined with the demand weights (in the case of equal all the preference weights), as well as the ordered median solution concept defined with the preference weights (in the case of equal all the demand weights). This paper studies basic models and properties of the weighted ordered median problem (WOMP) taking into account the demand weights following the WOWA aggregation rules. Linear programming formulations were introduced for optimization of the WOWA objective with monotonic preference weights thus representing the equitable preferences in the WOMP. We show MILP models for general WOWA optimization.     

Permuting matrices to avoid forbidden submatrices

This paper attaches a frame to a natural class of combinatorial problems and points out that this class includes many important special cases.

A matrix M is said to avoid a set of matrices if M does not contain any element of as (ordered) submatrix. For a fixed set of matrices, we consider the problem of deciding whether the rows and columns of a matrix can be permuted in such a way that the resulting matrix M avoids all matrices in .

We survey several known and new results on the algorithmic complexity of this problem, mostly dealing with (0,1)-matrices. Among others, we will prove that the problem is polynomial time solvable for many sets containing a single, small matrix and we will exhibit some example sets for which the problem is NP-complete.     

The ordered capacitated facility location problem

In this paper, we analyze flexible models for capacitated discrete location problems with setup costs. We introduce a major extension with regards to standard models which consists of distinguishing three different points of view of a location problem in a logistics system. We develop mathematical programming formulations for these models using discrete ordered objective functions with some new features. We report on the computational behavior of these formulations tested on a randomly generated battery of instances.     

Portfolio construction based on stochastic dominance and target return distributions

Mean-risk models have been widely used in portfolio optimization. However, such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties. The problem is multi-objective and is transformed into a single objective problem by using the reference point method, in which target levels, known as aspiration points, are specified for the objective functions. A model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return. This concept is extended by additionally specifying reservation points, which act pre-emptively in the optimization model. The theoretical properties of the models are studied. The performance of the models on real data drawn from the Hang Seng index is also investigated.     

An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems

We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems.