STANDARD METHODS FOR CONSTRAINED OPTIMIZATION

Jan A. Snyman () and Daniel N. Wilke ()
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Jan A. Snyman: University of Pretoria
Daniel N. Wilke: University of Pretoria

Chapter Chapter 3 in Practical Mathematical Optimization, 2018, pp 71-112 from Springer

Abstract: Abstract Consider the general constrained optimization problem: $$\begin{aligned} {\mathop {{{\mathrm{minimize\,}}}}_\mathbf{x}}&f(\mathbf{x})\nonumber \\ \text {such that }&g_j(\mathbf{x})\le 0\ \ j=1,2,\dots , m\\&h_j(\mathbf{x})=0\ \ j=1,2,\dots , r.\nonumber \end{aligned}$$ The most simple and straightforward approach to handling constrained problems of the above form is to apply a suitable unconstrained optimization algorithm to a penalty function formulation of constrained problem. Unfortunately the penalty function method becomes unstable and inefficient for very large penalty parameter values if high accuracy is required. A remedy to this situation is to apply the penalty function method to a sequence of sub-problems, starting with moderate penalty parameter values, and successively increasing their values for the sub-problems. Alternatively, the Lagrangian function with associated necessary Karush-Kuhn-Tucker (KKT) conditions and duality serve to solve constrained problems that has led to the development of the Sequential Quadratic Programming (SQP) method that applies $$\text {Newton's}$$ method to solve the KKT conditions.

Date: 2018
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DOI: 10.1007/978-3-319-77586-9_3

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