Page 253 - Numerical Methods for Chemical Engineering

242 5 Numerical optimization

Because the constraints of (5.123) are linear, it is easier to identify how far one may move
before violating them. Quadratic problems such as (5.123) thus can be solved efficiently
with specialized techniques. For more on this topic (interior point, active set methods),
consult Nocedal & Wright (1999) or view the documentation for fmincon.

Constrained minimizer fmincon in MATLAB

The fmincon constrained minimizer
fmincon solves constrained optimization problems with the structure

minimize F(x)
subject to the constraints (linear and nonlinear)
(5.124)
Ax ≤ b A (eq) x = b (eq) c(x) ≤ 0 c (eq) (x) = 0

and the upper and lower bounds LB j ≤ x j ≤ UB j
The routine is called with the syntax

[x, F, exitflag, output, grad, Hessian] = fmincon (. . .
fun, x0, A, b, A eq, b eq, LB, UB, nonlcon, OPTIONS, P1, P2,...);
fun is the name of a function that returns the cost function (and optionally the gradient and
Hessian),
function F = fun(x, P1, P2, ...);

x0 is the initial guess, and A, b, A eq, and b eq specify the linear constraints (if any). The
nonlinear constraint functions are returned by a user-supplied routine,

function [c,c eq] = nonlcon(x, P1, P2, ...);
If any constraint is not present, use[]asaplaceholder. The vectors of lower and upper
bounds for each parameter are LB and UB.If x j has no lower bound, set LB(j) = -Inf.If x j
has no upper bound, set UB(j) = Inf. OPTIONS is set by optimset. P1, P2,...are optional
parameters to be passed to fun.
The output variables are the constrained local minimum x, the cost function value F,
exitflag and output that provide information around the work done by fmincon.

Example. Finding the closest points on two ellipses

We revisit the problem of finding the closest points on two ellipses by minimizing the
squared distance between them,

2
F(x 1 , y 1 , x 2 , y 2 ) = (x 1 − x 2 ) + (y 1 − y 2 ) 2 (5.125)