Criteria for Unconstrained Global Optimization

We develop criteria for the existence and uniqueness of the global minima of a continuous bounded function on a noncompact set. Special attention is given to the problem of parameter estimation via minimization of the sum of squares in nonlinear regression and maximum likelihood. Definitions of local convexity and unimodality are given using the level set. A fundamental theorem of nonconvex optimization is formulated: If a function approaches the minimal limiting value at the boundary of the optimization domain from below and its Hessian matrix is positive definite at the point where the gradient vanishes, then the function has a unique minimum. It is shown that the local convexity level of the sum of squares is equal to the minimal squared radius of the regression curvature. A new multimodal function is introduced, the decomposition function, which can be represented as the composition of a convex function and a nonlinear function from the argument space to a space of larger dimension. Several general global criteria based on majorization and minorization functions are formulated.     

Genetic algorithms and cellular automata in aquifer management

A groundwater management problem is presented involving pumping cost minimization with both well discharges and well locations as decision variables. A grid of candidate well locations is set up and optimal arrangements of wells are sought within this discrete space. A genetic algorithm approach is presented with the following particular features: (a) A suitable scaling is applied to the objective function in order to alleviate its regionally flat behavior. (b) No penalty functions are involved in constraint handling. Instead, the feasible region is transformed into a rectangular domain. The transformation introduced is proved to be bijective. (c) A binary representation of well configurations is presented and compared to a combinatorial one. The binary representation necessitates the introduction of specially designed genetic operators. Besides purely genetic algorithms, the concept of cellular automaton is introduced as the basis of an alternative formulation of the optimization problem. The lattice of the cellular automaton provides the discrete set of candidate well positions. The well configuration is represented by a group of agents occupying an equal number of lattice sites. The agents change positions as dictated by the structure of the automaton and, also, by an associated genetic algorithm, which directs the evolution of the whole scheme toward an optimal configuration. An improved performance of this approach is noted and discussed in comparison to the purely genetic algorithm schemes of the present work. A simulated annealing approach is also applied to the same problem for comparison purposes. Finally, a new and more efficient hybrid annealing–genetic approach is introduced and discussed.     

Effective Relaxations and Partitioning Schemes for Solving Water Distribution Network Design Problems to Global Optimality

In this paper, we address the development of a global optimization procedure for the problem of designing a water distribution network, including the case of expanding an already existing system, that satisfies specified flow demands at stated pressure head requirements. The proposed approach significantly improves upon a previous method of Sherali et al. (1998) by way of adopting tighter polyhedral relaxations, and more effective partitioning strategies in concert with a maximal spanning tree-based branching variable selection procedure. Computational experience on three standard test problems from the literature is provided to evaluate the proposed procedure. For all these problems, proven global optimal solutions within a tolerance of 10^–4% and/or within 1$ of optimality are obtained. In particular, the two larger instances of the Hanoi and the New York test networks are solved to global optimality for the very first time in the literature. A new real network design test problem based on the Town of Blacksburg Water Distribution System is also offered to be included in the available library of test cases, and related computational results are presented.     

Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound

We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper, we specialize the algorithm to the box-constrained case and study its implementation, which is shown to be a state-of-the-art method for globally solving box-constrained nonconvex quadratic programs. S. Burer was supported in part by NSF Grants CCR-0203426 and CCF-0545514.     

Information theory of optimization in chromatography: Analytical structure of optimization

Summary The analytical roles of chromatographic variables (column length, etc.) can be soundly comprehended and compared in terms of the precision (Φ) of measurements and efficiency (ϑ) of analysis which are described as Shannon information and information flow, respectively. The φϑ plots of the optimization process and the information Φ transmitted by a single peak are useful to understand the analytical structure of optimization. Variables treated here are mobile phase composition (X), column length (L), mobile phase velocity rate (u), detection wavelength (λ) and plate number (N).     

The anticipative concept in warehouse optimization using simulation in an uncertain environment

The modern business environment is highly unpredictable. An anticipation approach in a real case study is presented to cope with such instability and minimize the total inventory cost without stock-outs occurring and inventory capacity being exceeded. The anticipation concept is performed using simulation models supported by inventory control algorithms on a selected sample of representative items. The inventory control algorithms include Silver–Meal, Part period balancing, Least-unit cost, and Fuzzy inventory control algorithm based on fuzzy stock-outs, highest inventory level and total cost. Transportation cost is explicitly defined as a discrete function of shipment size. The algorithms are tested on historic data. Simulation results are presented and the risk of accepting them as reliable is discussed. The process of simulation model implementation is briefly discussed to further validate the model and train order managers to use the simulation model in their order placement process.     

Supply chain optimization of petroleum organization under uncertainty in market demands and prices

A multi-period stochastic planning model has been developed and implemented for a supply chain network of a petroleum organization operating in an oil producing country under uncertain market conditions. The proposed supply chain network consists of all activities related to crude oil production, processing and distribution. Uncertainties were introduced in market demands and prices. A deterministic optimization model was first developed and tested. The impact of uncertainty on the supply chain was studied by performing a sensitivity analysis in which ±20% deviations were introduced in market demands and prices of different commodities. A stochastic formulation was then proposed, which is based on the two-stage problem with finite number of realizations. The proposed stochastic programming approach proved to be quite effective in developing resilient production plans in light of high degree of uncertainty in market conditions. The anticipated production plans have a considerably lower expected value of perfect information (EVPI). The main conclusion of this study is that for an oil producing country with oil processing capabilities, the impact of economic uncertainties may be tolerated by an appropriate balance between crude exports and processing capacities.     

Selection of adequate optimization criteria in chromatographic separations

Computer-assisted optimization of chromatographic separations is still a fruitful activity. In fact, advances in computerized data handling should make the application of systematic optimization strategies much easier. However, in most contemporary applications, the optimization criterion is not considered to be a key issue (Vanbel, J Pharm Biomed, 21:603–610, 1999). In this paper, an update of the importance of selecting adequate criteria in chromatographic separation is presented.     

Optimizing the source distribution in fluid mixing

A passive scalar is advected by a velocity field, with a nonuniform spatial source that maintains concentration inhomogeneities. For example, the scalar could be temperature with a source consisting of hot and cold spots, such that the mean temperature is constant. Which source distributions are best mixed by this velocity field? This question has a straightforward yet rich answer that is relevant to real mixing problems. We use a multiscale measure of steady-state enhancement to mixing and optimize it by a variational approach. We then solve the resulting Euler-Lagrange equation for a perturbed uniform flow and for simple cellular flows. The optimal source distributions have many broad features that are as expected: they avoid stagnation points, favour regions of fast flow, and their contours are aligned such that the flow blows hot spots onto cold and vice versa. However, the detailed structure varies widely with diffusivity and other problem parameters. Though these are model problems, the optimization procedure is simple enough to be adapted to more complex situations.     

Pareto optimal solutions of a cost–time trade-off bulk transportation problem

A cost–time trade-off bulk transportation problem with the objectives to minimize the total cost and duration of bulk transportation without according priorities to them is considered. The entire requirement of each destination is to be met from one source only; however a source can supply to any number of destinations subject to the availability of the commodity at it. Two new algorithms are provided to obtain the set of Pareto optimal solutions of this problem. This work extends and generalizes the work related to single-objective and prioritized two-objective bulk transportation problems done in the past while providing flexibility in decision making.