Appendix a Brief Review of Mathematical Optimization
Appendix A Brief Review of Mathematical Optimization Ecce signum (Behold the sign; see the proof) Optimization is a mathematical discipline which determines a “best” solution in a quantitatively well-defined sense. It applies to certain mathematically defined prob- lems in, e.g., science, engineering, mathematics, economics, and commerce. Opti- mization theory provides algorithms to solve well-structured optimization problems along with the analysis of those algorithms. This analysis includes necessary and sufficient conditions for the existence of optimal solutions. Optimization problems are expressed in terms of variables (degrees of freedom) and the domain; objective function1 to be optimized; and, possibly, constraints. In unconstrained optimization, the optimum value is sought of an objective function of several variables without any constraints. If, in addition, constraints (equations, inequalities) are present, we have a constrained optimization problem. Often, the optimum value of such a problem is found on the boundary of the region defined by inequality constraints. In many cases, the domain X of the unknown variables x is X = IR n. For such problems the reader is referred to Fletcher (1987), Gill et al. (1981), and Dennis & Schnabel (1983). Problems in which the domain X is a discrete set, e.g., X = IN o r X = ZZ, belong to the field of discrete optimization ; cf. Nemhauser & Wolsey (1988). Dis- crete variables might be used, e.g., to select different physical laws (for instance, different limb-darkening laws) or models which cannot be smoothly mapped to each other. In what follows, we will only consider continuous domains, i.e., X = IR n. In this appendix we assume that the reader is familiar with the basic concepts of calculus and linear algebra and the standard nomenclature of mathematics and theoretical physics.
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