Part 4 Nonlinear Programming 4 3 Successive Linear

Part 4 Nonlinear Programming 4. 3 Successive Linear Programming

Basic Concept

Approach 1: Direct Use of Linear Programs The simplest and most direct use of the linearization construction is to replace the general nonlinear problem with a complete linearization of all problem functions at some selected estimate solution. The linearized problem takes the form of a linear program and can be solved as such.

Case 1. 1 The linearly constrained case

Case 1. 1 The approximate problem

Bounded Line Search

Frank-Wolfe Algorithm

Remark

Case 1. 2 The general LP case

Direct Linear Approximation

Remark In order attain convergence to the true optimum, it is sufficient that at each iteration an improvement be made in both the objective function and constraint infeasibility. This type of monotonic behavior will occur if the problem functions are mildly nonlinear.

Approach 2 Separable Programming The motivation for this technique stems from the observation that a good way of improving the linear approximation over a large interval is to partition the interval into subintervals and construct individual linear approximation over each subinterval, i. e. , piecewise linear approximation.

Case 2. 1 Single-Variable Functions

Line Segment in Interval k

Line Segment in Interval k

General Formula

Case 2. 2 Multivariable Separable Functions

General Formula

General Formula

Restricted Basis Entry Prior to entering one lambda into the basis (which will make it nonzero), a check should be made to ensure that no more than one other lambda associated with the same x_i is in the basis. If there is one such lambda in the basis, it has to be adjacent.

Example

k 1 0 0 0 2 1 1 -2 3 2 16 -8 4 3 81 -18

Homework