The Complexity of Minimum Ratio Spanning Tree Problems

We examine the complexity of two minimum spanning tree problems with rational objective functions. We show that the Minimum Ratio Spanning Tree problem is NP-hard when the denominator is unrestricted in sign, thereby sharpening a previous complexity result. We then consider an extension of this problem where the objective function is the sum of two linear ratios whose numerators and denominators are strictly positive. This problem is shown to be NP-hard as well. We conclude with some results characterizing sufficient conditions for a globally optimal solution.     

Optimal paths in bi-attribute networks with fractional cost functions

An important routing problem is to determine an optimal path through a multi-attribute network which minimizes a cost function of path attributes. In this paper, we study an optimal path problem in a bi-attribute network where the cost function for path evaluation is fractional. The problem can be equivalently formulated as the “bi-attribute rational path problem” which is known to be NP-complete. We develop an exact approach to find an optimal simple path through the network when arc attributes are non-negative. The approach uses some path preference structures and elimination techniques to discard, from further consideration, those (partial) paths that cannot be parts of an optimal path. Our extensive computational results demonstrate that the proposed method can find optimal paths for large networks in very attractive times.     

Structural properties of optimal schedules with preemption

Scheduling problems with preemption are considered, where each operation can be interrupted and resumed later without any penalty. We investigate some basic properties of their optimal solutions. When does an optimal schedule exist (provided that there are feasible schedules)? When does it have a finite/polynomial number of interruptions? Do they occur at integral/rational points only? These theoretical questions are also of practical interest, since structural properties can be used to reduce the search space in a practical scheduling application. In this paper we answer some of these basic questions for a rather general scheduling model (including, as the special cases, the classicalmodels such as parallelmachine scheduling, shop scheduling, and resource constrained project scheduling) and for a large variety of the objective functions including nearly all known. For some two special cases of objective functions (including, however, all classical ones), we prove the existence of an optimal solution with a special “rational structure.” An important consequence of this property is that the decision versions of these optimization scheduling problems belong to class NP.     

Normal forms for rational difference equations with applications to the global periodicity problem

We propose a classification and derive the associated normal forms for rational difference equations with complex coefficients. As an application, we study the global periodicity problem for second-order rational difference equations with complex coefficients. We find new necessary conditions as well as some new examples of globally periodic equations.     

Singular Fibers in Elliptic Fibrations on the Rational Elliptic Surface

We determine the combinations of singular fibers a locally holomorphic elliptic fibration on the rational elliptic surface can admit. This problem has been answered for globally holomorphic elliptic fibrations by Persson and Miranda [12], [9]; we compare our methods and results to theirs. In particular, we find combinations of singular fibers which can be realized by locally holomorphic fibrations but not by globally holomorphic ones.     

An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems

We propose an algorithm for the global optimization of continuous minimax problems involving polynomials. The method can be described as a discretization approach to the well known semi-infinite formulation of the problem. We proceed by approximating the infinite number of constraints using tools and techniques from semidefinite programming. We then show that, under appropriate conditions, the SDP approximation converges to the globally optimal solution of the problem. We also discuss the numerical performance of the method on some test problems. Financial support of EPSRC Grant GR/T02560/01 gratefully acknowledged.     

Portfolio optimization with linear and fixed transaction costs

We consider the problem of portfolio selection, with transaction costs and constraints on exposure to risk. Linear transaction costs, bounds on the variance of the return, and bounds on different shortfall probabilities are efficiently handled by convex optimization methods. For such problems, the globally optimal portfolio can be computed very rapidly. Portfolio optimization problems with transaction costs that include a fixed fee, or discount breakpoints, cannot be directly solved by convex optimization. We describe a relaxation method which yields an easily computable upper bound via convex optimization. We also describe a heuristic method for finding a suboptimal portfolio, which is based on solving a small number of convex optimization problems (and hence can be done efficiently). Thus, we produce a suboptimal solution, and also an upper bound on the optimal solution. Numerical experiments suggest that for practical problems the gap between the two is small, even for large problems involving hundreds of assets. The same approach can be used for related problems, such as that of tracking an index with a portfolio consisting of a small number of assets.     

Scatter Search—Wellsprings and Challenges

This article is concerned with two global optimization problems (P1) and (P2). Each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function, where at least one of the convex functions is a quadratic form. First, the article presents and validates a number of theoretical properties of these problems. Included among these properties is the result that, under a mild assumption, any globally optimal solution for problem (P1) must belong to the boundary of its feasible region. Also among these properties is a result that shows that problem (P2) can be reformulated as a convex maximization problem. Second, the article presents for the first time an algorithm for globally solving problem (P2). The algorithm is a branch and bound algorithm in which the main computational effort involves solving a sequence of convex programming problems. Convergence properties of the algorithm are presented, and computational issues that arise in implementing the algorithm are discussed. Preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem (P2), provided that the number of variables is not too large.     

A numerical method for an optimal control problem with minimum sensitivity on coefficient variation

In this paper, we consider a class of optimal control problem involving an impulsive systems in which some of its coefficients are subject to variation. We formulate this optimal control problem as a two-stage optimal control problem. We first formulate the optimal impulsive control problem with all its coefficients assigned to their nominal values. This becomes a standard optimal impulsive control problem and it can be solved by many existing optimal control computational techniques, such as the control parameterizations technique used in conjunction with the time scaling transform. The optimal control software package, MISER 3.3, is applicable. Then, we formulate the second optimal impulsive control problem, where the sensitivity of the variation of coefficients is minimized subject to an additional constraint indicating the allowable reduction in the optimal cost. The gradient formulae of the cost functional for the second optimal control problem are obtained. On this basis, a gradient-based computational method is established, and the optimal control software, MISER 3.3, can be applied. For illustration, two numerical examples are solved by using the proposed method.     

Dynamics and control of a system of two non‐interacting preys with common predator

In this paper, we consider a Holling type model, which describes the interaction between two preys with a common predator. First, we give some sufficient conditions for the globally asymptotic stability and prove that local stability implies global stability. Then, we present a set of sufficient conditions for the existence of a positive periodic solution with strictly positive components. Finally, the optimal control strategy is developed to minimize the number of predator and maximize the number of preys. We also show the existence of an optimal control for the optimal control problem and derive the optimality system. The technical tool used to determine the optimal strategy is the Pontryagin Maximum Principle. Finally, the numerical simulations of global stability and the optimal problem are given as the conclusion of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.     

On the existence of the stabilizing solution of generalized Riccati equations arising in zero-sum stochastic difference games: the time-varying case

In this paper, a large class of time-varying Riccati equations arising in stochastic dynamic games is considered. The problem of the existence and uniqueness of some globally defined solution, namely the bounded and stabilizing solution, is investigated. As an application of the obtained existence results, we address in a second step the problem of infinite-horizon zero-sum two players linear quadratic (LQ) dynamic game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such an optimal control problem, a crucial role is played by the unique bounded and stabilizing solution of the considered class of generalized Riccati equations.     

A hierarchical decomposition approach to retail shelf space management and assortment decisions

Shelf management is a crucial task in retailing. Because of the large number of products found in most retail stores (sometimes more than 60?000), current shelf space management models can only solve sub-problems of the overall store optimization problem, since the size of the complete optimization problem would be prohibitively large. Consequently, an optimal allocation of store shelf space to products has not yet been achieved. We show that a hierarchical decomposition technique, consisting of two interwoven models, is suitable to overcome this limitation and, thus, is capable of finding accurate solutions to very large and complex shelf space management problems. We further conclude that other important variables (such as product-price) can be included into the methodology and their optimal values can be determined using the same solution technique. Our methodology is illustrated on a real-life application where we predict a 22.33% increase in store profits if our model's solution is implemented.     

Lower complexity bounds for interpolation algorithms

We introduce and discuss a new computational model for the Hermite-Lagrange interpolation with nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known Hermite-Lagrange interpolation problems and algorithms. Like in traditional Hermite-Lagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit cost. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariski’s Main Theorem and the theory of places (or equivalently: valuations). We finish this paper by exhibiting two examples of Lagrange interpolation problems with nonlinear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants).In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and classes of interpolants to be nonlinear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using nonlinear techniques.We finish this paper highlighting the close connection of our complexity results in Hermite-Lagrange interpolation with a modern trend in software engineering: architecture tradeoff analysis methods (ATAM).     

An efficient deterministic optimization approach for rectangular packing problems

The rectangular packing problem aims to seek the best way of placing a given set of rectangular pieces within a large rectangle of minimal area. Such a problem is often constructed as a quadratic mixed-integer program. To find the global optimum of a rectangular packing problem, this study transforms the original problem as a mixed-integer linear programming problem by logarithmic transformations and an efficient piecewise linearization approach that uses a number of binary variables and constraints logarithmic in the number of piecewise line segments. The reformulated problem can be solved to obtain an optimal solution within a tolerable error. Numerical examples demonstrate the computational efficiency of the proposed method in globally solving rectangular packing problems.     

Composing local and global behaviors: Higher performance of spin glass based portfolio selection

The basic challenge in optimization is how to navigate through the many non-optimal and mediocre solutions toward the few globally optimal solutions, amidst the growing problem size and computation complexity. If the proximity to an optimal solution could be measured, a desirable technique could be one that navigates speedily, even if crudely, when an optimal solution is not likely to be next; and accurately, even if slowly, otherwise. In this paper, we propose a technique based on spin glass paradigm that uses the above heuristic to solve the classic portfolio selection problem. Study of spin glass paradigm reveals that limiting each spin's interactions to its local neighborhood increases the computational speed of the algorithm, but also introduces an error in performance measure. In contrast, extending each spin's reach globally provides an accurate measure of performance, but slows down the glass computations. Theoretical analysis reveals a decision threshold by which speedy versus accurate navigation, i.e. local versus global glass behavior, can be alternated. The resulting algorithm is then applied to five different world stock market portfolio selection problems consisting of Hang Seng, DAX 100, FTSE 100, S&P 100, and Nikkei. These results demonstrate utility of the hybrid local–global behavior and appropriateness of the proposed decision threshold. Specifically, the results of experiments show faster convergence without a significant loss of accuracy in reaching globally optimal solutions.     

A Bilinear Algorithm for Optimizing a Linear Function over the Efficient Set of a Multiple Objective Linear Programming Problem

The problem Q of optimizing a linear function over the efficient set of a multiple objective linear program serves several useful purposes in multiple criteria decision making. However, Q is in itself a difficult global optimization problem, whose local optima, frequently large in number, need not be globally optimal. Indeed, this is due to the fact that the feasible region of Q is, in general, a nonconvex set. In this paper we present a monotonically increasing algorithm that finds an exact, globally-optimal solution for Q. Our approach does not require any hypothesis on the boundedness of neither the efficient set EP nor the optimal objective value. The proposed algorithm relies on a simplified disjoint bilinear program that can be solved through the use of well-known specifically designed methods within nonconvex optimization. The algorithm has been implemented in C and preliminary numerical results are reported.     

Cutting ellipses from area-minimizing rectangles

A set of ellipses, with given semi-major and semi-minor axes, is to be cut from a rectangular design plate, while minimizing the area of the design rectangle. The design plate is subject to lower and upper bounds of its widths and lengths; the ellipses are free of any orientation restrictions. We present new mathematical programming formulations for this ellipse cutting problem. The key idea in the developed non-convex nonlinear programming models is to use separating hyperlines to ensure the ellipses do not overlap with each other. For small number of ellipses we compute feasible points which are globally optimal subject to the finite arithmetic of the global solvers at hand. However, for more than 14 ellipses none of the local or global NLP solvers available in GAMS can even compute a feasible point. Therefore, we develop polylithic approaches, in which the ellipses are added sequentially in a strip-packing fashion to the rectangle restricted in width, but unrestricted in length. The rectangle’s area is minimized in each step in a greedy fashion. The sequence in which we add the ellipses is random; this adds some GRASP flavor to our approach. The polylithic algorithms allow us to compute good, near optimal solutions for up to 100 ellipses.     

Annealing a Genetic Algorithm for Constrained Optimization

In this paper, we adapt a genetic algorithm for constrained optimization problems. We use a dynamic penalty approach along with some form of annealing, thus forcing the search to concentrate on feasible solutions as the algorithm progresses. We suggest two different general-purpose methods for guaranteeing convergence to a globally optimal (feasible) solution, neither of which makes any assumptions on the structure of the optimization problem. The former involves modifying the GA evolution operators to yield a Boltzmann-type distribution on populations. The latter incorporates a dynamic penalty along with a slow annealing of acceptance probabilities. We prove that, with probability one, both of these methods will converge to a globally optimal feasible state.     

DAG reversal is NP-complete

Runs of numerical computer programs can be visualized as directed acyclic graphs (DAGs). We consider the problem of restoring the intermediate values computed by such a program (the vertices in the DAG) in reverse order for a given upper bound on the available memory. The minimization of the associated computational cost in terms of the number of performed arithmetic operations is shown to be NP-complete. The reversal of the data-flow finds application, for example, in the efficient evaluation of adjoint numerical programs. We derive special cases of numerical programs that require the intermediate values exactly in reverse order, thus establishing the NP-completeness of the optimal adjoint computation problem. Last but not least we review some state-of-the-art approaches to efficient data-flow reversal taken by existing software tools for automatic differentiation.