Optimization Concepts and Applications in Engineering

Introduction

The majority of engineering problems involve constrained minimization – that is, the task is to minimize a function subject to constraints. A very common instance of a constrained optimization problem arises in finding the minimum weight design of a structure subject to constraints on stress and deflection. Important concepts pertaining to linear constrained problems were discussed in Sections 4.1–4.5 in Chapter 4 including active or binding constraints, Lagrange multipliers, computer solutions, and geometric concepts. These concepts are also relevant to nonlinear problems. The numerical techniques presented here directly tackle the nonlinear constraints, most of which call an LP solver within the iterative loop. In contrast, penalty function techniques transform the constrained problem into a sequence of unconstrained problems as discussed in Chapter 6. In this chapter, we first present graphical solution for two variable problems and solution using EXCEL SOLVER and MATLAB. Subsequently, formulating problems in “standard NLP” form is discussed followed by optimality conditions, geometric concepts, and convexity. Four gradient-based numerical methods applicable to problems with differentiable functions are presented in detail:

Each of these methods is accompanied by a computer program in the disk at the end of the book. The reader can learn the theory and application of optimization with the help of the software.