Zelda B. Zabinsky

Improving Hit-and-Run for global optimization

Improving Hit-and-Run is a random search algorithm for global optimization that at each iteration generates a candidate point for improvement that is uniformly distributed along a randomly chosen direction within the feasible region. The candidate point is accepted as the next iterate if it offers an improvement over the current iterate. We show that for positive definite quadratic programs, the expected number of function evaluations needed to arbitrarily well approximate the optimal solution is at most O(n5/2) wheren is the dimension of the problem. Improving Hit-and-Run when applied to global optimization problems can therefore be expected to converge polynomially fast as it approaches the global optimum.

Pure adaptive search in global optimization

Pure adaptive seach iteratively constructs a sequence of interior points uniformly distributed within the corresponding sequence of nested improving regions of the feasible space. That is, at any iteration, the next point in the sequence is uniformly distributed over the region of feasible space containing all points that are strictly superior in value to the previous points in the sequence. The complexity of this algorithm is measured by the expected number of iterations required to achieve a given accuracy of solution. We show that for global mathematical programs satisfying the Lipschitz condition, its complexity increases at mostlinearly in the dimension of the problem.

Pure adaptive search in Monte Carlo optimization

Pure adaptive search constructs a sequence of points uniformly distributed within a corresponding sequence of nested regions of the feasible space. At any stage, the next point in the sequence is chosen uniformly distributed over the region of feasible space containing all points that are equal or superior in value to the previous points in the sequence. We show that for convex programs the number of iterations required to achieve a given accuracy of solution increases at most linearly in the dimension of the problem. This compares to exponential growth in iterations required for pure random search.

An algorithm for scheduling a chemical processing tank line

Abstract This paper presents a model and an algorithm for scheduling a system in which parts are processed through a chemical processing tank line. The tank line is equipped with one piece of material-handling equipment. The tank line is modelled with a mixed integer linear programming formulation. The formulation is then used to develop a heuristic algorithm. The algorithm generates the optimum or near optimum schedule and is easy to apply in practice where no defectives are permitted.

Stochastic Methods for Practical Global Optimization

Engineering design problems often involve global optimization of functions that are supplied as ‘black box’ functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.

Optimal design of large composite panels with varying loads

Abstract This paper presents an optimization formulation for the design of large composite panels when loads vary over the panel. A methodology termed “blending” is introduced and used to ensure that a panel is manufacturable. Two ways of specifying the blending rules in optimal design formulation are set forth and compared. A global optimization algorithm, Improving Hit-and-Run (IHR), is used to find optimal designs. A composite panel is designed with and without using blending rules to demonstrate their effectiveness. The resulting designs show that blending rules are a great assistance in designing large composite panels that are tailored for varying loads in a practical manner.

Social Welfare Maximization in Transmission Enhancement Considering Network Congestion

Transmission enhancement in a competitive electricity market is analyzed in terms of its impact on the social welfare. A formulation of social welfare and the utility functions are derived in this research. Variations of supply and demand during a year are considered in transmission enhancement decisions. The proposed social welfare optimization problem is solved to determine the transmission congestion status under the optimal condition of the existing system. Transmission enhancement options are compared quantitatively to demonstrate their ability to increase social welfare, which provides valuable information for long-term transmission. The proposed method provides a systematic and quantitative tool for social welfare evaluation that is not available in industry today.

New reflection generator for simulated annealing in mixed-integer/continuous global optimization

To reduce the well-known jamming problem in global optimization algorithms, we propose a new generator for the simulated annealing algorithm based on the idea of reflection. Furthermore, we give conditions under which the sequence of points generated by this simulated annealing algorithm converges in probability to the global optimum for mixed-integer/continuous global optimization problems. Finally, we present numerical results on some artificial test problems as well as on a composite structural design problem.

Comparative Assessment of Algorithms and Software for Global Optimization

The thorough evaluation of optimization algorithms and software demands devotion, time, (code development and hardware) resources, in addition to professional objectivity. This general remark is particularly valid with respect to global optimization (GO) software since GO literally encompasses “all” mathematical programming models. It is easy not only to fabricate very challenging test problems, but also to find realistic GO problems that pose a formidable task for any algorithm of today and of tomorrow.A report on computational experiments should ideally cover a large number of aspects: a detailed description and practical background of the models; earlier related work; solution approaches; algorithm implementations and their parameterization; hardware platforms, operating systems, and software environments; an exact description of all performance measures; report of successes and failures; analysis of solver parameterization effects; statistical characteristics for randomized problem-classes; and a summary of results (in text, tabular and/or graphical forms).An extensive inventory of classical NLP and GO test problems, as well as more recent (and often much harder) test suites have been suggested. This paper reviews several prominent test collections, discusses comparison issues, and presents illustrative numerical results. A second paper will perform a comparative study using ideas presented here, drawing also on discussions at the Stochastic Global Optimization Workshop (held in New Zealand, June 2001).

Designing laminated composites using random search techniques

A computer program called UWCODA is presented. UWCODA is intended to assist in the design, analysis and optimization of composite plates. UWCODA combines a state-of-the-art global optimization algorithm (Improving Hit and Run) with classical lamination theory. Optimization results are presented for simple loading conditions as well as for complex, biaxial load conditions. The computer code proved to be very effective in the design of composite plates.