Inequality ConstraintsInequality Constraints

Lecture 7Lecture 7

Inequality Contraints (I)Inequality Contraints (I)

A Review of Lagrange MultipliersA Review of Lagrange Multipliers– As we discussed last time, the first order As we discussed last time, the first order

necessary conditions for a constrained optimum necessary conditions for a constrained optimum include that the projected gradient has to equal include that the projected gradient has to equal to zeroto zero

Z f xx' ( ) 0

Inequality Contraints (II)Inequality Contraints (II)

– An alternative condition based on lagrange An alternative condition based on lagrange multipliers ismultipliers is

x f x A( ) '

Inequality Contraints (III)Inequality Contraints (III)

– Linking the lagrange and the null spaceLinking the lagrange and the null space

L f x b a x b a xL f x a ax x

( ) ( ' ) ( ' )( ) ' '

1 1 1 2 2 2

1 1 2 2 0

x f x a aA

( ) ' ''

1 1 2 2

Inequality Contraints (IV)Inequality Contraints (IV)

– A numerical example:A numerical example:» In the constrained optimization problem which we In the constrained optimization problem which we

discussed in the last lecture, we derived an optimal discussed in the last lecture, we derived an optimal solution ofsolution of

x

..

.

.

1161310666

3950859546

Inequality Contraints (V)Inequality Contraints (V)

» The constraint matrix for the problem wasThe constraint matrix for the problem was

A

8199 11366 6 298 8 01410 10 10 10. . . .. . . .

Inequality Contraints (V)Inequality Contraints (V)

» The Lagrangian multiplier condition for this The Lagrangian multiplier condition for this problem then becomesproblem then becomes

x f x( )

.

...

. .. .. .. .

*

437 712388 069467 511440 613

8199 1011366 106 298 108 014 10

1

2

Inequality Contraints (VI)Inequality Contraints (VI)

» Solving for Solving for 11 and and 22 then involves solving for the then involves solving for the reduced form of the linear system:reduced form of the linear system:

8199 10 436 71211366 10 388 0696 298 10 467 5118 014 10 440 613

. . .. . .. . .. . .

Inequality Contraints (VII)Inequality Contraints (VII)

» Which yields a solution ofWhich yields a solution of

1 0 156750 1 566 2120 0 00 0 0

..

Inequality Contraints (VIII)Inequality Contraints (VIII)

– In a maximization problem:In a maximization problem:» If the constraint cuts the frontier below the global If the constraint cuts the frontier below the global

maximum the lagrange multiplier will take on a maximum the lagrange multiplier will take on a positive value implying that an increase in the right positive value implying that an increase in the right hand side of the constraint will increase the objective hand side of the constraint will increase the objective function value.function value.

» Similarly, if the constraint cuts the frontier above the Similarly, if the constraint cuts the frontier above the global maximum the lagrange multiplier will take on a global maximum the lagrange multiplier will take on a negative value implying that an increase in the right negative value implying that an increase in the right hand side of the constraint will cause the objective hand side of the constraint will cause the objective function to decline.function to decline.

Inequality Contraints (IX)Inequality Contraints (IX)

– Thus, in the portfolio problem, income is being Thus, in the portfolio problem, income is being constrained below its optimum.constrained below its optimum.

In general, the optimality conditions for a In general, the optimality conditions for a inequality contrained optimum are little inequality contrained optimum are little different than the conditions for an equality different than the conditions for an equality constrained optimum. The primary constrained optimum. The primary difference involves restricting the sign of difference involves restricting the sign of the Lagrange multipliers.the Lagrange multipliers.

Inequality Contraints (IX)Inequality Contraints (IX)

– Conditions for an inequality constrained Conditions for an inequality constrained maximummaximum

Ax bZ f xx' ( ) 0 x f x A( ) 'or

0

Z f x Zxx' ( )2is negative semidefinite

Inequality Contraints (IX)Inequality Contraints (IX)

– Given the same conditions, we see that the Given the same conditions, we see that the solution to the previous problem is not solution to the previous problem is not optimum since optimum since 22 = -15.675 (the conditions for = -15.675 (the conditions for a greater than constraint under minimization are a greater than constraint under minimization are the same as the conditions for less than the same as the conditions for less than constraints under maximization except that constraints under maximization except that under minimization the projected Hessian must under minimization the projected Hessian must be positive definite.be positive definite.

Inequality Contraints (X)Inequality Contraints (X)

– Taking a new problem from the Taking a new problem from the Constrained2.ma notebook, assume that we Constrained2.ma notebook, assume that we want to maximize utility defined aswant to maximize utility defined as

max

. . .. . . .. . . .

. . . .x

xxxx

x x x x

xxxx

4 3 1 9

1 025 0125 0166025 75 0005 0010125 0005 25 045

0166 001 045 125

1

2

3

4

1 2 3 4

1

2

3

4

3 2 4 9

1

2

3

4

xxxx

Y

Inequality Contraints (XI)Inequality Contraints (XI)

– We see that the unconstrained problem has a We see that the unconstrained problem has a global maximum in the positive quadrant. The global maximum in the positive quadrant. The exact maximumexact maximum

x

xxxx

1

2

3

4

191157194001145741352385

.

.

.

.

The income required to reach this optimum vectorThe income required to reach this optimum vectoris 33.06is 33.06

Inequality Contraints (XII)Inequality Contraints (XII)

– Thus, we want to evaluate two scenarios. Thus, we want to evaluate two scenarios. Under the first scenario the level of income is Under the first scenario the level of income is set at 40.0 which is beyond the level required set at 40.0 which is beyond the level required for the global maximum.for the global maximum.

» Under this scenario we solveUnder this scenario we solve

max ( )xf x

st x3 2 4 5 40

Inequality Contraints (XIII)Inequality Contraints (XIII)

which yields an optimum ofwhich yields an optimum of

x

2 139642 129812 610043 77626

.

...

and an optimal and an optimal of -.14383 which does not satisfy of -.14383 which does not satisfythe condition for an optimum that the condition for an optimum that > 0. > 0.

Inequality Contraints (XIV)Inequality Contraints (XIV)

– The second scenario then involves constraining The second scenario then involves constraining y to be less than 30.y to be less than 30.

» Under this scenario, the optimal solution becomesUnder this scenario, the optimal solution becomes

x

1810841856180 94317341237

.

.

.

.With a With a of 0.065537. Thus, the Lagrange of 0.065537. Thus, the Lagrangemultiplier condition is metmultiplier condition is met