Optimization Methods- Historical Development
The existence of optimization methods can be traced to the days of Newton, Lagrange, and Cauchy. The foundations of calculus of variations, were laid by Bernoulli, Euler and Lagrange. The method of optimization for constrained problems, became known by the name of its inventor, Lagrange. Cauchy made the first application of the steepest descent method to solve unconstrained optimization problems. Development of the simplex method by Dantzig in 1947 for linear programming problems. Work by Kuhn and Tucker in 1951 on the necessary and sufficient conditions for the optimal solution of programming problems laid the foundation for later research in non-linear programming
The enunciation of the principle of optimality in 1957 by Bellman for dynamic programming problems. The contributions of Zoutendijk and Rosen to nonlinear programming during the early 1960s have been very significant.
• Geometric programming was developed in the 1960s by Duffin, Zener, and Peterson.
The necessity to optimize more than one objective or goal while satisfying the physical limitations led to the development of multi-objective programming methods. Goal programming is a well-known technique for solving specific types of multi-objective optimization problems and proposed by Charnes and Cooper in1961. High-speed digital computers made implementation of the complex optimization procedures and encouraged further research on newer methods.
Engineering applications of optimization
Design of civil engineering structures such as frames, foundations, bridges, towers, chimneys and dams for minimum cost. Design of water resources systems for obtaining maximum benefit. Design of optimum pipeline networks for process industry. Design of aircraft and aerospace structure for minimum weight. Finding the optimal trajectories of space vehicles. Optimum design of electrical machinery such as motors, generators and transformers.
Optimum design of control systems
Optimum design of distribution systems
Optimum design of load scheduling
Optimum design of transmission systems
Design Variables: Select parameters which are highly sensitive to the proper working of design and these are called design variables
Thumb rule is to choose few design variables
Constraints: There are two types i) Inequality type ii) Equality type
Objective function
It can be Maximization or Minimization
Choose most important objective as Objective function
Other objectives can be included in the form of constraints
If more than one objective is considered, it is called Multi- Objective Optimization
Classification of Optimization Algorithms
Single variable optimization algorithms
i) Direct methods- Do not use derivative information
ii) Gradient methods- Use derivative information of Objective function
Multi variable optimization algorithms
i) Direct
ii) Gradient
Constrained optimization algorithms
These are mostly used in engineering optimization
Specialized optimization algorithms
Integer programming
Geometric programming
Nontraditional optimization algorithms
Genetic Algorithm
PSO
Differential Evolution etc.
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