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Concept Version 7
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Optimization

Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives.

Learning Objective

  • Define optimization as finding the maxima and minima for a function, and describe its real-life applications


Key Points

    • Many design problems can also be expressed as optimization programs.
    • Optimization relies heavily on finding maxima and minima. For this, calculus is useful.
    • An example would be companies seeking to maximize sales while minimizing costs.

Terms

  • stochastic

    Random, randomly determined

  • optimization

    the design and operation of a system or process to make it as good as possible in some defined sense


Full Text

Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives. Optimization process that involves only a single variable is rather straightforward. After finding out the function $f(x)$ to be optimized, local maxima or minima at critical points can be easily found. (Of course, end points may have maximum/minimum values as well.) The same strategy applies for optimization with several variables.

Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the engineering optimization, and another recent and growing subset of this field is multidisciplinary design optimization, which, while useful in many problems, has in particular been applied to aerospace engineering problems.

Economics is closely linked to optimization of agents. Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibria. In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem, are economic optimization problems. Insofar as they behave consistently, consumers are assumed to maximize their utility, while firms are usually assumed to maximize their profit. Also, agents are often modeled as being risk-averse, thereby preferring to avoid risk. Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization. Trade theory also uses optimization to explain trade patterns between nations. The optimization of market portfolios is an example of multi-objective optimization in economics.

Another field that uses optimization techniques extensively is operations research. Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses stochastic programming to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and stochastic optimization methods.

Maxima

Finding maxima is useful in optimization problems.

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