Lecture18

Chapter 12

Optimization with inequality

constraints

Here we want to solve the following constrained maximization problem:

Maximize f(x)

subject togj(x) 0, j = 1, . . . , m, and x2X,

whereXis a non-empty subset ofRn_andf,gj_{(·),j = 1, . . . , m, are functions}

fromX toR. Here the constraint set is given by

C={x2X :gj(x) 0, j = 1, . . . , m}.

A point x⇤ _{2 X is a constrained local maximum if there exists an open}

ball B✏(x⇤) ⇢ Rn _{such that f(x}⇤_{) f(x) for all x 2 B✏(x}⇤_{)\C; x}⇤ _{is a}

constrained global maximum if it solves the problem above. A constrained minimization problem and its local and global minima can be defined anal-ogously.

12.1

Saddle point

Definition 12.1 (Saddle point). A pair (x⇤, ⇤)2X⇥Rn

+ is a saddle point if

(x, ⇤)_ (x⇤, ⇤)_ (x⇤, )

for all x2X and 2Rn

+, where

(x, ) = f(x) + g(x).

92CHAPTER 12. OPTIMIZATION WITH INEQUALITY CONSTRAINTS

12.1.1

Constrained global maximum and saddle points

The following result shows the equivalence of a constrained global maximum and a saddle point.

Theorem 12.1. If (x⇤_, ⇤_)2X⇥Rm

+ is a saddle point, then

• ⇤_g(x⇤_{) = 0;} • g(x⇤) 0; and

• x⇤ _{is a point of constrained global maximum.}

Proof. By the second inequality in the definition of a saddle point, for all

2Rn

+

(x⇤, ) =f(x⇤) + g(x⇤) (x⇤, ⇤)

=f(x⇤) + ⇤g(x⇤) or, g(x⇤) ⇤g(x⇤)

or, ( ⇤)g(x⇤) 0.

Pick any j 2 {1, . . . , m}, let j = j⇤ + 1 and i = ⇤i for all i 6= j. This

gives gj_(x⇤_{) 0. Repeating this procedure, we obtain g}j_(x⇤_{) 0 for all}

j = 1, . . . , m. Consequently, ⇤_·g(x⇤_{) 0. On the other hand, setting = 0}

above, we get ⇤ ·g(x⇤)0. Therefore, ⇤·g(x⇤) = 0. Now, for all x2C, f(x)_f(x) + ⇤g(x)

f(x⇤) + ⇤g(x⇤) =f(x⇤).

The first inequality holds since ⇤ 2 Rn

+ and gj(x) 0, j = 1, . . . , m. The second inequality holds by the first inequality in the definition of a saddle point. The equality holds because ⇤_g(x⇤_{) = 0.}

A converse of this result also holds as follows.

Theorem 12.2. Suppose x⇤ _{2 X is a constrained global maximum where}

X is convex,f and gj_{s, j = 1, . . . , m, are concave functions, and there exists}

¯

x2X such thatgj_{(¯x)>0,j = 1, . . . , m(Slater’s condition). There exists a} ⇤ _2Rn

+ such that

• ⇤_g(x⇤_{) = 0, and}

• (x⇤_, ⇤_{) is a saddle point.}

12.2

Kuhn-Tucker Conditions and Saddle Points

LetX _⇢Rn _{be open and f, g}j_{,j = 1, . . . , m be C}1 _{functions from X to R.} A pair (x⇤, ⇤)2X⇥Rm

+ satisfies the Kuhn-Tucker conditions if

• @@xfi(x

⇤_{) +}Pm j=1 ⇤j@g

j @xi(x

⇤_{) = 0, i= 1, . . . , n and}

• g(x⇤_{) 0 and} ⇤_g(x⇤_{) = 0.}

The condition ⇤_g(x⇤_{) = 0 is known as the complementary slackness}

condi-tion. It is equivalent to ⇤_jgj_(x⇤_{) = 0 for each j = 1, . . . , m. Hence it implies}

that if gj(x⇤) > 0 then j⇤ = 0 but if gj(x⇤) = 0 then ⇤j can be either zero

or strictly positive.

The following result uses the fact that if X be an open convex set and f, gj, j = 1, . . . , m are C1 and concave functions from X to R, then any (x⇤_, ⇤_{) 2 X ⇥ R}m

+ satisfying the Kuhn-Tucker conditions is maximizing (x, ⇤) =f(x) + ⇤g(x). This fact and the complementary slackness condi-tions imply that (x⇤, ⇤) must be a saddle point.

Theorem 12.3. LetX be an open convex set and let f,gj_{, j = 1, . . . , m be}

C1 _{and concave functions fromX toR. If a pair (x}⇤_, ⇤_)2X⇥Rm

+ satisfies the Kuhn-Tucker conditions then

• (x⇤_, ⇤_{) is a saddle point and}

• x⇤ is a point of constrained global maximum.

The converse is also easy to prove and left as an exercise.

Exercise 12.2.1. Let X be an open set and let f, gj_{, j = 1, . . . , m be C}1 functions fromX toR. If a pair (x⇤_, ⇤_)2X⇥Rm

+ is a saddle point then it satisfies the Kuhn-Tucker conditions.

12.3

Necessary and su

ffi

cient conditions for

constrained local maximum

Now we show that if x⇤ 2 X is a constrained local maximum, then under suitable conditions, there exists ⇤ 2 Rm+ such that (x⇤, ⇤) satisfies the Kuhn-Tucker conditions.

Theorem 12.4. Let X _⇢ Rn _{be open and f, g}j _{be C}1 _{functions from} X to R. Suppose x⇤ 2 X is a constrained local maximum of f subject to the k inequality constraints gj_{(x)  b}

94CHAPTER 12. OPTIMIZATION WITH INEQUALITY CONSTRAINTS

generality, suppose that the first k0 of these constraints are binding and the remainingk k0 constraints are not binding. Further suppose that the Jacobian matrix corresponding to thek0 binding constraints has full rankk0. Form the Lagrangian

L(x, ) =f(x)

k

X

j=1

j(gj(x) bj).

Then there exist multipliers ⇤

1, . . . , ⇤ksatisfying the Kuhn-Tucker conditions,

i.e.,

• @L

@xi(x

⇤_, ⇤_{) = 0, i= 1, . . . , n;}

• ⇤

j(gj(x⇤, ⇤) bj) = 0, j = 1, . . . , k;

• ⇤

j 0,j = 1, . . . , k;

• gj_(x⇤_{) b}

j 0, j = 1, . . . , k.

Proof. Since the gj_{s are continuous functions, there exists an open ball B}

around x⇤ _{such that for all x 2 B, g}j_{(x) < b}

j for j = k0 + 1, . . . , k. Note that x⇤ _{maximizes f on B for the constraint set {x 2 B : g}j_{(x) = b}

j, j =

1, . . . , k0}(If there was anotherx⇤⇤ 2Bfor whichfhad a higher value on this constraint set, then this would give a higher value for the original constrained maximization problem in B, contradicting that x⇤ _{is a local maximum).}

Since this is a local maximum subject to equality constraints, and the non-degenerate constraint qualification corresponding to this problem is satisfied (the Jacobian matrix corresponding to these constraints has full rank), there existµ⇤₁, . . . , µ⇤_k0 such that

• @Lˆ

@xi(x

⇤_{, µ}⇤_{) = 0, i= 1, . . . , n;}

• gj_⇤(x⇤_{) =b}

j, j = 1, . . . , k0,

where ˆLx, µ=f(x) Pk0

j=1µj(gj(x) bj)).

Now consider the Lagrangian for the original problem, L(x, ) =f(x) Pk

j=1 j(gj(x) bj). Set ⇤j = µ⇤j for j = 1, . . . , k0 and ⇤j = 0 for j =

k0+ 1, . . . , k. Then observe that (x⇤_, ⇤_{) satisfies all Kuhn-Tucker conditions}

except one, ⇤_j 0, j = 1, . . . , k0. We now prove that this condition is also satisfied.

Consider the system of k0 equations gj_{(x) = b}

j, j = 1, . . . , k0 in k0 +n

small, we can find x such that g1_{(x) = b1 t and g}j_{(x) = b}

j, j = 2, . . . , k0.

This implies that there exists aC1 _{curvex(t) fort2[0,✏) such thatx(0) =x}⇤

and g1_{(x(t)) = b1 t and g}j_{(x(t)) =b}

j, j = 2, . . . , k0. Let v=x0(0). By the

chain rule, Dg1_(x⇤_{)v= 1 and Dg}j_(x⇤_{)v= 0, j = 2, . . . , k0. Since x(t) lies}

in the constraint set for all t ₂ [0,✏) and x⇤ _{maximizes f in the constraint}

set, f must be nonincreasing along x(t). Therefore,

d

dtf(x(t)) _t=0 =Df(x

⇤_)v0.

LetDxL(x⇤) be the derivative ofL(x, ) evaluated at x⇤. Sincex⇤ is a local

maximum of f subject to the k equality constraints,

0 =DxL(x⇤)v

=Df(x⇤)v

k

X

j=1

jDgj(x⇤)v

=Df(x⇤)v 1Dg1(x⇤)v =Df(x⇤)v+ 1.

Since Df(x⇤_{)v  0,}

1 0. Similarly, we can show that j 0 for j =

2, . . . , k0.

An analogous statement can be proven for Kuhn-Tucker conditions be-ing necessary for constrained local minimum. It must be noted that the inequality constraints here are of the form gj_{(x) b}

j.

Exercise 12.3.1. State and prove the version of Theorem 12.4 when some of the constraints are equality constraints.

The Kuhn-Tucker conditions are also sufficient for a local maximum pro-vided the relevant second order condition is satisfied: supposeg representsk inequality constraints andh representsm equality constraints and let and µ be the corresponding multipliers. Suppose (x⇤_, ⇤_{, µ}⇤_{) satisfy the}

Kuhn-Tucker conditions (with mixed constraints). Further, the Hessian matrix of the Lagrangian with respect toxat (x⇤_, ⇤_{, µ}⇤_{) is negative definite at the}

96CHAPTER 12. OPTIMIZATION WITH INEQUALITY CONSTRAINTS

An analogous statement can be proven for Kuhn-Tucker conditions being sufficient for constrained local minimum provided the relevant second order condition is satisfied. This second order condition is that the Hessian matrix of the Lagrangian with respect to x at (x⇤, ⇤, µ⇤) is positive definite at the linear constraint set _{v ₆= 0 : Dgk0(x⇤)v = 0, Dh(x⇤)v = 0} where gk0 represent the binding inequality constraints. It must be noted that the inequality constraints here are of the form gj_{(x) b}

j. This holds when the