Single and Multiple Objective Optimization

Single and Multiple Objective Optimization covers a wide range of techniques for optimizing functions. Single objective optimization examines functions and identifies the function minima and maxima over a predetermined range. Single objective optimization is often handled using differential calculus. However, in many applications, obtaining extrema using the methods of calculus is intractable. In these cases alternate techniques are required. Many artificial intelligence techniques are based on identifying a global minimum or maximum when multiple extrema exist. Evolutionary algorithms such as genetic algorithms, differential evolution, and particle swarm optimization were developed as modern alternatives to the standard optimization approaches such as differentiation and Newton's method. These evolutionary techniques are examined and compared against a suite of test functions to measure performance of each technique under a variety of operating conditions. Multiobjective optimization does not typically result in a single optimum value. Instead, a set of incomparable points is identified on the Pareto frontier. These points are not comparable to each other, but are superior to other potential solutions. Although a single operating point is not identified, the optimal value must be among the points in the Pareto frontier. Techniques for identifying the Pareto frontier are examined and tested using a suite of test problems. Some of the techniques examined are the weighted sum method, Normal-Boundary Intersection (NBI), Normal Constraint, Strength Pareto Evolutionary Algorithm (SPEA2), Nondominated Sorting Genetic Algorithm (NSGA2), and Directed Search Domain (DSD). These techniques may be compared in terms of the number of function evaluations, the distribution of points on the frontier, the number of frontier points identified, along with many other performance measures.