Goldstein Steplength Criterion
F ( x) + 2g T p
F ( x + p )
F ( x) + g p
F ( x) + 1g T p
Gamma Steplength Criterion
F ( x + p )
g T p
g T p
F ( x) + g T p
2. LINEAR CONSTRAINTS
In this chapter, we consider optimization problems with constraints defined by linear functions of the variables. The general form of a linear function is v(x) = aTx , for some row
vector aT and scalar . By linearity, the column vect
Supplementary Notes on Linear Programming
This note is concerned with barrier-function methods for the solution of linear programs in the standard form
Ax = b,
where A is an m n matrix with m n.
1.1. METHODS FOR UNIVARIATE FUNCTIONS
1.1.1. Finding the Zero of a Univariate Function
It will be shown that a necessary condition for x to be an unconstrained minimizer of a
twice-continuously differentiable univariate function f (x) is that f 0 (x ) = 0
RESULTS FROM CALCULUS AND LINEAR ALGEBRA
Several useful results from real analysis and basic calculus are summarized here.
Limit point of a sequence in <n . Let cfw_xk be a sequence in <n . A point x is
called a limit point of cfw_xk if there e
An inequality-constrained nonlinear programming problem may be posed in
subject to c(x) 0,
where f (x) is a nonlinear function and c(x) is an m-vector of nonlinear
functions with ith component ci (x), i = 1,. .
Figure 4a. The peinte an, 5:; end :3 denote the ret three membere ef the eequenee
generated by the bieeeen algal-item applied he e en the intehrel [e, e]. Figure 4b. The paint n+1 denetee the Newton approximation to the zero of re].
The new point in dened
1.2. OPTIMALITY CONDITIONS FOR UNCONSTRAINED PROBLEMS
We shall be concerned with the unconstrained problem:
where F (x) is a real-valued function defined on an open convex set D <n .
Definition 1.2.1. A point x 2 <n is a local minimize
Even one nonlinear constraint considerably increases the difficulty of solving an
optimization problem. It often pays to try and eliminate nonlinear constraints if at
all possible. A measure of the increase in difficulty ma