This paper addresses the solution of bound-constrained optimization problems using algorithms that require only the availability of objective function values but no derivative information. We refer to these algorithms as derivative-free algorithms. Fueled by a growing number of applications in science and engineering, the development of derivativefree optimization algorithms has long been studied, and it has found renewed interest in recent time. Along with many derivative-free algorithms, many software implementations have also appeared. The paper presents a review of derivative-free algorithms, followed by a systematic comparison of 22 related implementations using a test set of 502 problems. The test bed includes convex and nonconvex problems, smooth as well as nonsmooth problems. The algorithms were tested under the same conditions and ranked under several criteria, including their ability to find near-global solutions for nonconvex problems, improve a given starting point, and refine a near-optimal solution. A total of 112,448 problem instances were solved. We find that the ability of all these solvers to obtain good solutions diminishes with increasing problem size. For the problems used in this study, TOMLAB/MULTI-MIN, TOMLAB/GLCCLUSTER, MCS and TOMLAB/LGO are better, on average, than other derivative-free solvers in terms of solution quality within 2,500 function evaluations. These global solvers outperform local solvers even for convex problems. Finally, TOMLAB/OQNLP, NEWUOA, and TOMLAB/MULTIMIN show superior performance in terms of refining a nearoptimal solution.

Empirical and theoretical studies of preference structures of investors have long shown that personal and corporate utility is typically multimodal, implying that the same investor can be risk-averse at certain levels of wealth while risk-seeking at others. In this paper, we consider the problem of optimizing the portfolio of an investor with an indefinite quadratic utility function. The convex and concave segments of this utility reflect the investor's attitude towards risk, which changes based on deviations from a fixed goal. Uncertainty is modeled via a finite set of scenarios for the returns of securities. A global optimization approach is developed to solve the proposed nonconvex optimization problem. We present computational results which investigate the effect of short sales and demonstrate that the proposed approach systematically produces portfolios with higher values of skewness than the classical expectation-variance approach.

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