Optimization problems with a single variable have special importance in optimization theory. The reason for this is that multidimensional optimization problems are usually solved iteratively through optimization algorithms that utilize single variable optimization approaches. Typically, at the kth iteration of an algorithm, a promising search direction s(k) 2 <n in the n-dimensional parameter space is first determined. Starting from the solution at the kth iteration x(k), a line search is carried out in the direction s(k). A general point along that line is given by x 1/4 x(k) þ as(k), where a > 0 is the search parameter. The optimal value of the search parameter is the one that minimizes the objective function.

Chapter Contents:

• 4.1 Introduction
• 4.2 Bracketing approaches
• 4.2.1 Fixed line search
• 4.2.2 Accelerated line search
• 4.3 Derivative-free line search
• 4.3.1 Dichotomous line search
• 4.3.2 The interval-halving method
• 4.3.3 The Fibonacci search
• 4.3.4 The Golden Section method
• 4.4 Interpolation approaches
• 4.4.1 Quadratic models
• 4.4.2 Cubic interpolation
• 4.5 Derivative-based approaches
• 4.5.1 The classical Newton method
• 4.5.2 A quasi-Newton method
• 4.5.3 The Secant method
• 4.6 Inexact line search
• A4.1 Tuning of electric circuits
• A4.1.1 Tuning of a current source
• A4.1.2 Coupling of nanowires
• A4.1.3 Matching of microwave antennas
• References
• Problems

Inspec keywords: optimisation; search problems

Other keywords: n-dimensional parameter space; one-dimensional optimization-line search; search parameter; multidimensional optimization problems; single variable optimization approach; objective function minimization

Subjects: Optimisation techniques; Optimisation techniques