Multi-parametric Optimization and Control. Efstratios N. Pistikopoulos
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strict inequality holds in expression (1.4) for
1.1.1.2 Properties of Convex Functions
1 Let be convex functions defined on a convex subset . Their summation(1.5) is convex, and if at least of one is a strictly convex function, then their summation is strictly convex.
2 Let a be a positive number and be a (strictly) convex function defined in a convex subset . Then the product is (strictly) convex.
3 Let be a (strictly) convex function defined in , and be an increasing convex function defined on the range of in . Then, the composite function defined in is a (strictly) convex function.
4 Let be convex functions defined on a convex subset . If these functions are bounded from above, their pointwise supremum(1.6) is a convex function on .
5 Let be concave functions defined on a convex subset . If these functions are bounded from below, their pointwise infimum(1.7) is a concave function on .
1.1.2 Optimality Conditions
We introduce the following definitions for the solution of general nonlinear optimization problems:
Definition 1.6 (Local Minimum)
(1.8)
Definition 1.7 (Global Minimum)
(1.9)
A constrained nonlinear optimization problem, which aims to minimize a real valued function
(1.10)
Problem (1.10) is a nonlinear optimization problem, if and only if, at least one of
Definition 1.8 (Active Constraints)
An inequality constraint
Remark 1.1
If one step of the dual simplex algorithm consists of changing one element of the active set, i.e. let
The first‐order constraint qualifications that will be presented in the following text are necessary prerequisites to identify whether a feasible point
Linear independence constraint qualification: The gradients for all and for all are linearly independent.
Slater constraint qualification: The constraints for all are pseudo‐convex1 at , while the constraints for all are quasi‐convex or quasi‐concave.2 In addition, the gradients are linearly independent and there exists such that and .
1.1.2.1 Karush–Kuhn–Tucker Necessary Optimality Conditions
Let
(1.11)
These conditions are the Karush–Kuhn–Tucker (KKT) Necessary Conditions and they are the basis for the solution of nonlinear optimization problems.
1.1.2.2 Karun–Kush–Tucker First‐Order Sufficient Optimality Conditions
Consider the sets
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