Efstratios N. Pistikopoulos - Multi-parametric Optimization and Control

is called a global minimum if

(1.9)

A constrained nonlinear optimization problem, which aims to minimize a real valued function subject to the inequality constraints and equality constraints is denoted as

(1.10)

Problem ( 1.10) is a nonlinear optimization problem, if and only if, at least one of is a nonlinear function. We assume that the aforementioned functions are continuous and differentiable.

An inequality constraint is called active at a point if . Conversely, is called inactive if .

If one step of the dual simplex algorithm consists of changing one element of the active set, i.e. let , then the dual pivot involving the constraint yields .

The first‐order constraint qualifications that will be presented in the following text are necessary prerequisites to identify whether a feasible point is a local optimum of the function .

Linear independence constraint qualification: The gradients for all and for all are linearly independent.

Slater constraint qualification: The constraints for all are pseudo‐convex 1at , while the constraints for all are quasi‐convex or quasi‐concave. 2In addition, the gradients are linearly independent and there exists such that and .

Let and be differentiable at a feasible solution , and let have continuous partial derivatives at . In addition, let be the number of active inequality constraints at . Then if one of the aforementioned constraint qualifications hold, there exist Lagrange multipliers such that

(1.11)

These conditions are the Karush–Kuhn–Tucker (KKT) Necessary Conditions and they are the basis for the solution of nonlinear optimization problems.

Consider the sets and . Then, if the following conditions hold:

is pseudo‐convex at with respect to all other feasible points x.

for all are quasi‐convex at with respect to all other feasible points x.

for all are quasi‐convex at with respect to all other feasible points x.

for all are quasi‐concave at with respect to all other feasible points x.

then is a global optimum of problem ( 1.10). If the aforementioned conditions hold only within a ball of radius around , then is a local optimum of problem ( 1.10).

Consider the following problem:

(1.12)