constraint qualification

Lecture notes based onFoundations of Mathematical Economics

c⃝ 2001 Michael CarterAll rights reserved

Constraint qualification and sufficientconditions

1 Constraint qualification

We derived the Kuhn-Tucker theorem by assuming regularity of the bind-ing constraints. With the possibility of corner solutions, regularity is toostringent. The regularity condition will not necessarily apply throughoutthe constraint set. Verifying the regularity condition requires knowing theoptimal solution, but the first-order conditions will not identify the optimalsolution unless the regularity condition is satisfied – “Catch 22”.To illustrate, consider the problem

max𝑥1,𝑥2

𝑥1

subject to 𝑥2 − (1 − 𝑥1)3 ≤ 0

−𝑥2 ≤ 0

𝑥1

𝑥2

𝐷𝑔1(𝑥)

𝐷𝑔2(𝑥)

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The solution is obviously (1, 0), but this does not satisfy the Kuhn-Tuckerconditions. The gradients of the two constraints at (1, 0) are (0, 1) and (0,−1)which are linearly dependent. The constraints are not regular at the optimum(1, 0). The problem is that the set

𝐿(x∗) = {dx ∈ 𝑋 : ∇𝑔𝑗(x∗)𝑇dx ≤ 0, 𝑗 = 1, 2} = {(𝑥1, 0)}

does not represent the set of feasible perturbations 𝑇 (x∗) = {(𝑥1, 0) : 𝑥1 ≤1}. If we add an additional constraint 𝑔3(𝑥1, 𝑥2) = 𝑥1 + 𝑥2 ≤ 1, the threegradients are still linearly dependent, but now

𝐿(x∗) = {dx ∈ 𝑋 : ∇𝑔𝑗(x∗)𝑇dx ≤ 0, 𝑗 = 1, 2, 3} = 𝑇 (x∗)

and the optimum (1, 0) satisfies the Kuhn-Tucker conditions. We seek condi-tions on the constraints g such that the set 𝐿(x∗) correctly represents the setof feasible perturbations 𝑇 (x∗), which is known as the cone of tangents.

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1.1 Constraint qualification conditions (Theorem 5.4)

Suppose that x∗ is a local solution of

maxx∈𝑋

𝑓(x) subject to 𝑔(x) ≤ 0

at which the binding constraints 𝐵(x∗) satisfy one of the following constraintqualification conditions.

Concave CQ 𝑔𝑗 is concave for every 𝑗 ∈ 𝐵(x∗)

Pseudoconvex CQ 𝑔𝑗 is pseudoconvex and there exists x̂ ∈ 𝑋 such that𝑔𝑗(x̂) < 0 for every 𝑗 ∈ 𝐵(x∗)

Quasiconvex CQ 𝑔𝑗 is quasiconvex, ∇𝑔𝑗(x∗) ∕= 0 and there exists x̂ ∈ 𝑋

such that 𝑔𝑗(x̂) < 0 for every 𝑗 ∈ 𝐵(x∗)

Regularity The set { ∇𝑔𝑗(x∗) : 𝑗 ∈ 𝐵(x∗) } is linearly independent

Then the Kuhn-Tucker conditions are necessary for an optimal solution.

The most common constraint qualification conditions encountered are

∙ 𝑔𝑗 linear =⇒ 𝑔𝑗 concave (e.g. linear programming)

∙ Slater condition: 𝑔𝑗 convex and there exists x̂ ∈ 𝑋 such that 𝑔𝑗(x̂) <0 for every 𝑗

1.2 Constraint qualification with nonnegative variables (Corollary 5.4.1)

Provided the binding constraints 𝑗 ∈ 𝐵(x∗) in the problem

maxx≥0

𝑓(x) subject to g(x) ≤ 0

satisfy any one of the following constraint qualification conditions:

Concave CQ

Pseudoconvex CQ

Quasiconvex CQ

the Kuhn-Tucker conditions are necessary for an optimal solution.Proof. The problem can be specified as

maxx

𝑓(x)

subject to g(x) ≤ 0h(x) = −x ≤ 0

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We note that h is linear, and is therefore both concave and convex. Further𝐷h[x] ∕= 0 for every x. Therefore, if g satisfies one of the three constraintqualification conditions, so does the combined constraint (g,h). By Theorem5.4, the Kuhn-Tucker conditions are necessary for a local optimum. □

Example 5.41: The consumer’s problem The consumer’s problem

maxx≥0

𝑢(x)

subject to p𝑇x ≤ 𝑚

has one functional constraint 𝑔(x) = p𝑇x ≤ 𝑚 and 𝑛 inequality constraintsℎ𝑖(x) = −𝑥𝑖 ≤ 0, the gradients of which are

∇𝑔 = p ∇ℎ𝑖 = e𝑖, 𝑖 = 1, 2, . . . , 𝑛

where e𝑖 is the 𝑖 unit vector (Example 1.79). Provided all prices are positivep > 0, it is clear that these are linearly independent and the regularitycondition of Corollary 5.3.2 is always satisfied. However, it is easier to appealdirectly to Corollary 5.4.1, and observe that the budget constraint 𝑔(x) =p𝑇x ≤ 𝑚 is linear and therefore concave.

2 Sufficient conditions

We know that the KT conditions are sufficient when 𝑓 is concave and 𝑔 isconvex. The following is a signficant generalization (e.g. consumer theory).

2.1 Sufficient conditions for a global optimum (Theorem 5.5)

Suppose that x∗ satisfies the KT conditions and

∙ 𝑓 is pseudoconcave

∙ 𝑔 is quasiconvex

Then x∗ is a global maximum.Proof. For every 𝑗

either 𝜆𝑗 = 0 which implies 𝜆𝑗∇𝑔𝑗(x∗)𝑇 (x− x∗) = 0 for every x ∈ 𝑋

or 𝑔𝑗(x∗) = 0 and therefore 𝑔𝑗(x) ≤ 0 = 𝑔𝑗(x

∗) for every x ∈ 𝐺 = {x ∈ 𝑋 :𝑔𝑗(x) ≤ 0, 𝑗 = 1, 2, . . . , 𝑚}. Quasiconvexity implies

∇𝑔𝑗(x∗)𝑇 (x− x∗) ≤ 0 for every x ∈ 𝐺

and since 𝜆𝑗 ≥ 0

𝜆𝑗∇𝑔𝑗(x∗)𝑇 (x− x∗) ≤ 0 for every x ∈ 𝐺

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The KT condition is

∇𝑓(x∗) =

𝑚∑𝑗=1

𝜆𝑗∇𝑔𝑗(x∗)

and therefore

∇𝑓(x∗)𝑇 (x− x∗) =𝑚∑𝑗=1

𝜆𝑗∇𝑔𝑗(x∗)𝑇 (x− x∗) ≤ 0

Since 𝑓 is pseudoconcave, this implies that 𝑓(x∗) ≥ 𝑓(x) □2.2 Arrow-Enthoven theorem (Corollary 5.5.1)

Suppose that x∗ satisfies the KT conditions and

∙ 𝑓 is quasiconcave and ∇𝑓(x∗) ∕= 0∙ 𝑔 is quasiconvex

Then x∗ is a global maximum.

2.3 Necessary and sufficient conditions (Corollary 5.5.3)

Suppose that

∙ 𝑓 is quasiconcave and 𝑔 quasiconvex

∙ ∇𝑓(x∗) ∕= 0 and ∇𝑔𝑗(x∗) ∕= 0 for every 𝑗 ∈ 𝐵(x∗)

∙ there exists x̂ such that 𝑔𝑗(x̂) < 0 for every 𝑗 ∈ 𝐵(x∗)

Then the KT conditions are necessary and sufficient for a global optimum.

Example 5.43: The consumer’s problem Convex preferences and non-satiation imply 𝑢 quasiconcave and ∇𝑢(x) ∕= 0 for every x ∈ 𝑋 . Thereforethe KT conditions

∇𝑢(x∗) = 𝜆p

are necessary and sufficient for utility maximization.

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Example: Linear programming (Section 5.4.4) A linear program-ming problem is a special case of the general constrained optimization prob-lem

maxx≥0

𝑓(x) (1)

subject to 𝑔(x) ≤ 0

in which both the objective function 𝑓 and the constraint function g arelinear. Consequently, the Kuhn-Tucker conditions are both necessary andsufficient for a global optimum. The simplex algorithm is an efficient algo-rithm for solving the Kuhn-Tucker conditions.

3 Homework

1. Suppose that x∗ is a local solution of

maxx∈𝑋

𝑓(x) subject to 𝑔(x) ≤ 0

at which the binding constraints 𝐵(x∗) satisfy one of the above con-straint qualification conditions, so that x∗ satisfies the Kuhn-Tuckerconditions

∇𝑓(x∗) =

𝑚∑𝑗=1

𝜆𝑗∇𝑔𝑗(x∗) and 𝜆𝑗𝑔𝑗(x

∗) = 0 𝑗 = 1, 2 . . . , 𝑚

Show that the Lagrange multipliers 𝜆1, 𝜆2, . . . , 𝜆𝑚 are unique if andonly if 𝐵(x∗) satisfies the Regularity condition, that is { ∇𝑔𝑗(x

∗) : 𝑗 ∈𝐵(x∗) } is linearly independent.

2. Suppose that a firm has contracted with its union to hire at least 𝑙units of labour at rate 𝑤1 per unit. It can also hire non-union labourat 𝑤2 < 𝑤1 per hour. Assume that labour is the only input. Union andnon-union labour are equally productive, with diminishing marginalproduct. Output is sold at a fixed price 𝑝.

(a) Derive and interpret the first-order conditions for maximizing profit.(Note: Although the optimal solution is obvious, please show howit can be derived from the first-order conditions.)

(b) Are these conditions necessary for a solution?

(c) Are they sufficient to identify a global optimum?

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Solutions 5

1 Since 𝜆𝑗 = 0 for every 𝑗 /∈ 𝐵(x∗), the Kuhn-Tucker conditions imply

∇𝑓(x∗) =∑

𝑗∈𝐵(x∗)

𝜆𝑗∇𝑔𝑗(x∗)

If ∇𝑔𝑗(x∗), 𝑗 ∈ 𝐵(x∗) are independent, then the 𝜆𝑗 are unique (Exercise

1.137). Conversely, if there exist 𝜇1, 𝜇2, . . . , 𝜇𝑚 such that with 𝜇𝑗 ∕= 𝜆𝑗 forsome 𝑗 and

∇𝑓(x∗) =∑

𝑗∈𝐵(x∗)

𝜇𝑗∇𝑔𝑗(x∗)

then∇𝑓(x∗) −∇𝑓(x∗) =

∑𝑗∈𝐵(x∗)

(𝜆𝑗 − 𝜇𝑗)∇𝑔𝑗(x∗) = 0

which implies that ∇𝑔𝑗(x∗), 𝑗 ∈ 𝐵(x∗) are dependent (Exercise 1.133).

2 (a) Let 𝑥1 denote the quantity of union labour and 𝑥2 the quantity ofnon-union labour hired by the firm. The firm’s optimisation problemis

max𝑥1≥𝑙,𝑥2≥0

𝑝𝑓(𝑥1 + 𝑥2) − 𝑤1𝑥1 − 𝑤2𝑥2

which can be written as

max𝑥1 and 𝑥2≥0

𝑝𝑓(𝑥1 + 𝑥2) − 𝑤1𝑥1 − 𝑤2𝑥2

subject to 𝑔(𝑥1) = 𝑙 − 𝑥1 ≤ 0

Forming the Lagrangean

𝐿(𝑥1, 𝑥2, 𝜆) = 𝑝𝑓(𝑥1 + 𝑥2) − 𝑤1𝑥1 − 𝑤2𝑥2 − 𝜆(𝑙 − 𝑥1)

the Kuhn-Tucker conditions for an optimum are

𝐷𝑥1𝐿 = 𝑝𝑓 ′(𝑥1 + 𝑥2) − 𝑤1 + 𝜆 = 0 (1)

𝐷𝑥2𝐿 = 𝑝𝑓 ′(𝑥1 + 𝑥2) − 𝑤2 ≤ 0 𝑥2 ≥ 0(𝑝𝑓 ′(𝑥) − 𝑤2

)𝑥2 = 0 (2)

𝑥1 ≥ 𝑙 𝜆 ≥ 0 𝜆(𝑙 − 𝑥1) = 0

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There are two cases:

𝑥2 > 0 (1) and (2) imply

𝑤1 + 𝜆 = 𝑤2 =⇒ 𝜆 > 0 =⇒ 𝑥1 = 𝑙

and𝑝𝑓 ′(𝑥2 + 𝑙) = 𝑤2

𝑥2 = 0 (2) implies

𝑝𝑓 ′(𝑥1) ≤ 𝑤2 < 𝑤1 =⇒ 𝜆 > 0 =⇒ 𝑥1 = 𝑙

In both cases, the firm hires exactly 𝑙 units of union labour (𝑥1). If𝑝𝑓 ′(𝑙) > 𝑤2, then the firm hires additional units of non-union labour(𝑥2).

(b) Since the constraint function 𝑔(𝑥1) = 𝑙 − 𝑥1 is linear, it is satisfies theCQ condition. The Kuhn-Tucker conditions are necessary.

(c) Diminishing marginal product means that the production function 𝑓 isconcave. Therefore the objective function 𝑝𝑓(𝑥1 +𝑥2)−𝑤1𝑥1−𝑤2𝑥2 isconcave. Since the constraint function 𝑔 is convex (linear), the Kuhn-Tucker conditions are also sufficient.

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constraint qualification

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