Part 4 Nonlinear Programming 4.1 Introduction

Standard Form

An Intuitive Approach to Handle the Equality Constraints One method of handling just one or two equality constraints is to solve for 1 or 2 variables and eliminate them from problem formulation by substitution.

Use of Lagrange Multipliers to Handle n Equality Constraints and m+n Variables

Equivalent Formulation

Choice of Decision Variables For a given optimization problem, the choice of which variables to designate as the decision (control) variables is not unique. It is only a matter of convenience to make a distinction between decision and state variables.

1st Derivation of Necessary Conditions (i)

1st Derivation of Necessary Conditions (ii)

1st Derivation of Necessary Conditions (iii)

2nd Derivation of Necessary Conditions (i)

2nd Derivation of Necessary Conditions (ii)

2nd Derivation of Necessary Conditions (iii)

2nd Derivation of Necessary Conditions (iv)

2nd Derivation of Necessary Conditions - General Formulation

3rd Derivation with Lagrange Multipliers

Example: Solution:

Sensitivity Interpretation

Generalized Sensitivity

Problems with Inequality Constraints Only

 

Two Scenarios at Minimum *

  Area of improvement exists if Eq (14) is not satisfied.

  minimum  

J Inequality Constraints and N Variables

Geometrical Interpretation At any local constrained optimum, no (small) allowable change in the problem variables can improve the value of the objective function. lies within the cone formed by the negative gradients of the active constraints.

General Formulation

Active Constraints

Kuhn-Tucker Conditions

Kuhn-Tucker Necessity Theorem

Remarks The Kuhn-Tucker necessity theorem helps to identify points that are not optimal. If the KTC are satisfied, there is no assurance that the solution is truly optimal.

y1 y2

Sensitivity

Constraint Qualification When the constraint qualification is not met at the optimum, there may or may not exist a solution to the Kuhn-Tucker problem.

Second-Order Optimality Conditions

Necessary and Sufficient Conditions for Optimality If a Kuhn-Tucker point satisfies the second-order sufficient conditions, then optimality is guaranteed.

Basic Idea of Penalty Methods

Example

Exact L1 Penalty Function

Equivalent Smooth Constrained Problem

Barrier Method

Generalized Cases

Mixed Penalty-Barrier Method