in Nonlinear Systems —
An Interactive Tutorial
Preface
— Frank Lloyd Wright
Quantitative decision problems are frequently modeled by optimizing the value of a suitable objective function under given feasibility constraints. The objective function typically expresses overall system "performance," such as profit, utility, loss, risk, or error. The constraints originate from physical, technical, economic, logical — or possibly some other — considerations.
Although many such models naturally belong to the realm of "traditional" (local scope) continuous optimization — notably, to linear and convex programming — there exist also numerous cases in which the convexity requirements are not satisfied or are not simply verifiable. The essential reason for this fact is that the phenomena and processes modeled can be highly nonlinear. In case of a complex nonlinear system description, the associated decision model may — and frequently will — have multiple locally optimal solutions. In most realistic cases, the number of such local solutions is unknown, and the quality of local and global solutions may differ substantially. Therefore multiextremal decision models can be very difficult, and — as a rule — standard optimization strategies are not directly applicable to solve them.
The objective of global optimization (GO) is to determine the "absolutely best" solution and corresponding optimum value in (potentially) multiextremal models. Early attempts have been directed at GO at least since the late 1950s. Especially in the past two decades, model and algorithm development, and rapidly increasing computational power have led to significant advances. There exist dozens of textbooks and informative World Wide Web sites, a few professional journals, and literally thousands of research articles devoted to the subject.
This work presents a concise, practical introduction to models and algorithms that enable the analysis and solution of decision problems in presence of multiple optima. A special emphasis is placed upon computational aspects. First, a brief review of frequently used models and methods is provided, then GO software development and performance testing issues are highlighted. This is followed by a discussion of several interesting GO applications, arising in the sciences and engineering. The LGO integrated modeling and solver system is introduced and used to solve these examples: The corresponding demonstration program files are also made available. A list of references — mainly consisting of recent books and Web sites — assists the further orientation of the reader in this important and challenging field.
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