Notes | Mechatronic Systems and Laboratory

2.2 Problem statement

We shall consider the following NLP problem:

Definition 2.7

Nonlinear Program

\begin{matrix} \min_{x} & f (x) \\ s.t. & g (x) \leq 0 \\ h (x) = 0 \\ x \in X \end{matrix}

where $X$ is a subset of $ℝ^{n_{x}}$ , $x$ is a vector of $n_{x}$ components $x_{1}, . . ., x_{n}$ , and $f : X \to ℝ$ , $g : X \to ℝ^{n_{g}}$ and $h : X \to ℝ^{n_{h}}$ are defined on $X$ .

Remark 2.3

The function $f$ is usually called the objective function or criterion function or performance index. Each of the constraints $g_{i} (x) \leq 0$ , $i = 1, . . ., n_{g}$ , is called an inequality constraint, and each of the constraints $h_{i} (x) = 0$ , $i = 1, . . ., n_{h}$ , is called an equality constraint.

Remark 2.4

Note also that the set $X$ typically includes lower and upper bounds on the variables. The reason for separating variable bounds from the other inequality constraints is that they can play a useful role in some algorithms, i.e. they are handled in a specific way.

Remark 2.5

A vector $x \in X$ satisfying all the constraints is called a feasible solution to the problem. The collection of all such points forms the feasible region. The NLP problem, then, is to find a feasible point $(x ⋆$ such that $f (x) \geq f ((x ⋆)$ for each feasible point $(x$ .

Needless to say, a NLP problem can be stated as a maximization problem, and the inequality constraints can be written in the form $g (x) \geq 0$ .

Example 2.1. Consider the following problem:

\begin{matrix} \min_{x} & {(x_{1} - 2)}^{2} + {(x_{2} - 1)}^{2} \\ s.t. & x_{1}^{2} - x_{2} \leq 0 \\ x_{1} + x_{2} - 2 \leq 0 \\ - x_{1} - 1 \leq 0 \end{matrix}

The objective function and the three inequality constraints are:

\begin{matrix} f (x_{1}, x_{2}) & = & {(x_{1} - 2)}^{2} + {(x_{2} - 1)}^{2} \\ g_{1} (x_{1}, x_{2}) & = & x_{1}^{2} - x_{2} \\ g_{2} (x_{1}, x_{2}) & = & x_{2} + x_{1} - 2 \\ g_{3} (x_{1}, x_{2}) & = & - x_{1} - 1 \end{matrix}

Figure 2.3 illustrates the feasible region. The problem, then, is to find a point in the feasible region with the smallest possible value of ${(x_{1} - 2)}^{2} + {(x_{2} - 1)}^{2}$ . Note that points $(x_{1}, x_{2})$ with ${(x_{1} - 2)}^{2} + {(x_{2} - 1)}^{2} = c$ are circles with radius $c$ and centre $(2, 1)$ . This circle is called the contour of the objective function having the value $c$ . In order to minimize $c$ , we must find the circle with the smallest radius that intersects the feasible region. As shown in figure 2.3 , the smallest circle corresponds to $c = 1$ and intersects the feasible region at the point $(1, 1)$ . Hence, the optimal solution occurs at the point $(1, 1)$ and has an objective value equal to $1$ .

The graphical approach used in the above example, i.e. find an optimal solution by determining the objective contour with the smallest objective value that intersects the feasible region, is only suitable for small problems. It becomes intractable for problems containing contours of the objective function in more than three variables as well as for problems having complicated objective and/or constraint functions.

pict — Figure 2.3:: Geometric solution of a nonlinear problem.

This chapter is organized as follows:

we start by defining what is meant by optimality and give conditions under which a minimum (or a maximum) exists for a nonlinear program;
then, we discuss the special properties of convex programs;
then, both necessary and sufficient conditions of optimality are presented for NLP problems;
successively, we consider unconstrained problems, problems with inequality constraints, and problems with both equality and inequality constraints;
finally, several numerical optimization techniques will be presented, which are instrumental to solve a great variety of NLP problems.