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# Basic Results in Optimization

In this appendix, some basic results for deterministic optimization problems are presented. We first consider static optimization and present results for both the unconstrained and constrained cases. Following this, we look at dynamic optimization, where both multistage and continuous time formulations are considered.

## D.1 UNCONSTRAINED STATIC OPTIMIZATION

The simplest case is that in which x is a scalar decision variable to be selected optimally to maximize a scalar function L(x)1. Let x* denote the local optimal x. x* may or may not exist and when it does, there can one or several local or global maxima.

### Scalar x

If x is a real variable, then to be a local maximum, x* must satisfy the first order necessary condition given by

as well as the condition

Any solution to D1 yields a stationary point. A sufficient condition for x* to yield a local maximum is given by (D1) and

In general, it is necessary to solve (D1) computationally to obtain the stationary points and then check (D2) to determine which yield a local maximum. Many different techniques for accomplishing this have been developed. One of the simplest is the first order gradient ...

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