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Strong Duality - Nonlinear Programming - Lecture Slides, Slides of Computer Science

All India Institute of Medical Sciences Computer Science

These are the Lecture Slides of Nonlinear Programming which includes Convex Cost, Linear Constraints, Duality Theorem, Linear Programming Duality, Quadratic Programming Duality, Linear Inequality, Constrained Problem, Minimize, Feasible etc.Key important points are: Strong Duality, Linear Equality Constraints, Fenchel Duality, Lagrange Multiplier, Dual Problem, Maximize, Optimal Value, Finite, Vector, Lagrange Multiplier

Typology: Slides

2012/2013

Uploaded on 03/27/2013

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NONLINEAR PROGRAMMING

LECTURE 20: STRONG DUALITY

LECTURE OUTLINE

• Strong Duality Theorem

• Linear equality constraints. Fenchel Duality.

********************************

• Consider the problem

minimize f (x)

subject to x ∈ X, gj

(x) ≤ 0, j =1,...,r,

assuming −∞ <f

∗ < ∞.

• µ

∗ is a Lagrange multiplier if µ

∗ ≥ 0 and f∗ =

infx∈X L(x, µ

∗).

• Dual problem: Maximize q(µ)= infx∈X L(x, µ)

subject to µ ≥ 0.

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Download Strong Duality - Nonlinear Programming - Lecture Slides and more Slides Computer Science in PDF only on Docsity!

NONLINEAR PROGRAMMING

LECTURE 20: STRONG DUALITY

LECTURE OUTLINE

• Strong Duality Theorem

• Linear equality constraints. Fenchel Duality.

• Consider the problem

minimize f (x)

subject to x ∈ X, gj (x) ≤ 0 , j = 1,... , r,

assuming −∞ < f

∗ < ∞.

μ ∗

is a Lagrange multiplier if μ

∗

≥ 0 and f

∗

infx∈X L(x, μ ∗ ).

• Dual problem: Maximize q(μ) = infx∈X L(x, μ)

subject to μ ≥ 0.

DUALITY THEOREM FOR INEQUALITIES

• Assume that X is convex and the functions

f : n

→ , gj :

→ are convex over X. Fur-

thermore, the optimal value f

∗

is finite and there

exists a vector x¯ ∈ X such that

gj (¯x) < 0 , ∀ j = 1,... , r.

• Strong Duality Theorem: There exists at least

one Lagrange multiplier and there is no duality

gap.

(0,f*) (μ,1) w z A = {(z,w) | there is an x in X such that g(x) ≤ z, f(x) ≤ w } ( g(x),f(x) ) S =^ {(g(x),f(x)) | x^ ∈^ X}

′

LINEAR EQUALITY CONSTRAINTS

• Suppose we have the additional constraints

i e x − di = 0, i = 1,... , m

• We need the notion of the affine hull of a convex

set X [denoted af f (X)]. This is the intersection of

all hyperplanes containing X.

• The relative interior of X, denoted ri(X), is the set

of all x ∈ X s.t. there exists � > 0 with

z | ‖z − x‖ < �, z ∈ af f (X) ⊂ X,

that is, ri(X) is the interior of X relative to af f (X).

• Every nonempty convex set has a nonempty

relative interior.

′

DUALITY THEOREM FOR EQUALITIES

• Assumptions:

− The set X is convex and the functions f , gj

are convex over X.

− The optimal value f

∗

is finite and there exists

a vector ¯x ∈ ri(X) such that

gj (¯x) < 0 , j = 1,... , r, e i x¯ − di = 0, i = 1,... , m.

• Under the preceding assumptions there exists

at least one Lagrange multiplier and there is no

duality gap.

FENCHEL DUALITY FRAMEWORK

• Consider the problem

minimize f 1 (x) − f 2 (x)

subject to x ∈ X 1 ∩ X 2 ,

where f 1 and f 2 are real-valued functions on

and X 1 and X 2 are subsets of

• Assume that −∞ < f

∗

• Convert problem to

minimize f 1 (y) − f 2 (z)

subject to z = y, y ∈ X 1 , z ∈ X 2 ,

and dualize the constraint z = y.

q(λ) = inf f 1 (y) − f 2 (z) + (z − y) ′ λ y∈X 1 , z∈X 2 = inf z ′ λ − f 2 (z) − sup y ′ λ − f 1 (y) z∈X (^2) y∈X 1 = g 2 (λ) − g 1 (λ)

DUALITY THEOREM

0 0 X 1 f 1 (x) (λ,-1) (λ,-1) Slope = λ sup 2 (x) - x'λ} = - g 2 (λ) x ∈ X 2 f 2 (x) X 2 Slope = λ x x {f inf {f 1 (x) - x'λ} = - g 1 (λ) x ∈ X 1

Strong Duality - Nonlinear Programming - Lecture Slides, Slides of Computer Science

Related documents

Partial preview of the text

Download Strong Duality - Nonlinear Programming - Lecture Slides and more Slides Computer Science in PDF only on Docsity!

NONLINEAR PROGRAMMING

LECTURE 20: STRONG DUALITY

LECTURE OUTLINE

• Strong Duality Theorem

• Linear equality constraints. Fenchel Duality.

• Consider the problem

minimize f (x)

subject to x ∈ X, gj (x) ≤ 0 , j = 1,... , r,

assuming −∞ < f

is a Lagrange multiplier if μ

≥ 0 and f

∗

• Dual problem: Maximize q(μ) = infx∈X L(x, μ)

subject to μ ≥ 0.

DUALITY THEOREM FOR INEQUALITIES

• Assume that X is convex and the functions

→ , gj : 

→  are convex over X. Fur-

thermore, the optimal value f

is finite and there

exists a vector x¯ ∈ X such that

• Strong Duality Theorem: There exists at least

one Lagrange multiplier and there is no duality

gap.

LINEAR EQUALITY CONSTRAINTS

• Suppose we have the additional constraints

• We need the notion of the affine hull of a convex

set X [denoted af f (X)]. This is the intersection of

all hyperplanes containing X.

• The relative interior of X, denoted ri(X), is the set

of all x ∈ X s.t. there exists � > 0 with

that is, ri(X) is the interior of X relative to af f (X).

• Every nonempty convex set has a nonempty

relative interior.

DUALITY THEOREM FOR EQUALITIES

• Assumptions:

− The set X is convex and the functions f , gj

are convex over X.

− The optimal value f

is finite and there exists

a vector ¯x ∈ ri(X) such that

• Under the preceding assumptions there exists

at least one Lagrange multiplier and there is no

duality gap.

FENCHEL DUALITY FRAMEWORK

• Consider the problem

minimize f 1 (x) − f 2 (x)

subject to x ∈ X 1 ∩ X 2 ,

where f 1 and f 2 are real-valued functions on 

and X 1 and X 2 are subsets of 

• Assume that −∞ < f

• Convert problem to

minimize f 1 (y) − f 2 (z)

subject to z = y, y ∈ X 1 , z ∈ X 2 ,

and dualize the constraint z = y.

DUALITY THEOREM

• Assume that

− X 1 and X 2 are convex

− f 1 and f 2 are convex and concave over X 1

and X 2 , respectively

− The relative interiors of X 1 and X 2 intersect

• The duality theorem for equalities applies and

shows that

and that the maximum above is attained.

→ , gj :

→ are convex over X. Fur-

where f 1 and f 2 are real-valued functions on

and X 1 and X 2 are subsets of