KKT conditions I Lecture 14 ME EN 575 Andrew Ning aning@byu.edu - - PDF document

KKT conditions I

Lecture 14

ME EN 575 Andrew Ning aning@byu.edu

Outline

Equality Constraints

minimize f(x) with respect to x ∈ Rn subject to lb < x < ub ˆ cj(x) = 0, j = 1, . . . , ˆ m ck(x) ≤ 0, k = 1, . . . , m

Equality Constraints

Motivating Problem:

minimize x1 + x2 subject to x2

1 + x2 2 = 8

Small group exercise

• What is the unconstrained optimum?
• Draw the direction of the function gradient:

∇f.

• Rewrite the constraint in our convention.
• Draw the direction(s) for the constraint

gradients: ∇c.

• What is the constrained optimal solution and

where is it located?

• What do you notice about ∇f and ∇c at the
• ptimum? Does that make sense?
• How could you define the optimality criteria

mathematically?

A More Formal Motivation

Define the Lagrangian L(x, λ) = f(x) + λˆ c(x)

Extend to m constraints

∂L ∂xi = ∂f ∂xi +

ˆ m

• j=1

ˆ λj ∂ˆ cj ∂xi = 0, (i = 1, . . . , n) ∂L ∂ˆ λj = ˆ cj = 0, (j = 1, . . . , ˆ m).

λj is called a Lagrange multiplier, and there is a separate one for each constraint.