1.4 FOUNDATIONS OF CONSTRAINED OPTIMIZATION

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1 Essential Microeconomics -- 4 FOUNDATIONS OF CONSTRAINED OPTIMIZATION Fundamental Theorem of linear Programming 3 Non-linear optimization problems 6 Kuhn-Tucker necessary conditions Sufficient conditions 3 John Riley 4 October 202

2 Essential Microeconomics -2- Linear Optimization Step : We consider linear maimization problems (linear programming problems) Step 2: In the second step we then obtain necessary and sufficient conditions for general problems by eamining the linearized approimation of the original problem Linear maimization problem Ma{ q = a X n } where X = { 0, A z}, 0 A is anm nmatri We assume that X is bounded Thus by the etreme value theorem there eists a solution We also assume that X has a non-empty interior John Riley 4 October 202

3 Essential Microeconomics -3- Proposition 4-: Fundamental Theorem of Linear Programming Suppose that solves Ma{ a0 X n } where X = { 0, A z} is bounded and has a non-empty interior Then there eists a shadow price vector λ 0 such that a A, (00-) 0 λ 0 and the following complementary slackness conditions hold (i) ( a 0 λ A ) = 0 and (ii) λ ( z A ) = 0 Moreover, these conditions are not only necessary but sufficient That is, if A z, a0 λ A 0, and the complementary slackness conditions hold, then is the solution to the linear optimization problem ** John Riley 4 October 202

4 Essential Microeconomics -4- Proposition 4-: Fundamental Theorem of Linear Programming Suppose that solves Ma{ a0 X n } where X = { 0, A z} is bounded and has a non-empty interior Then there eists a shadow price vector λ 0 such that a A, (00-2) 0 λ 0 and the following complementary slackness conditions hold (i) ( a 0 λ A ) = 0 and (ii) λ ( z A ) = 0 Moreover, these conditions are not only necessary but sufficient That is, if A z, a0 λ A 0, and the complementary slackness conditions hold, then is the solution to the linear optimization problem Necessity follows directly from Proposition -4 That is, although we interpreted the linear model as an activity model of a firm with n plants, the formal analysis applies to any linear constrained optimization problem * John Riley 4 October 202

5 Essential Microeconomics -5- Proposition 4-: Fundamental Theorem of Linear Programming Suppose that solves Ma{ a0 X n } where X = { 0, A z} is bounded and has a non-empty interior Then there eists a shadow price vector λ 0 such that a A, (00-3) 0 λ 0 and the following complementary slackness conditions hold (i) ( a 0 λ A ) = 0 and (ii) λ ( z A ) = 0 Moreover, these conditions are not only necessary but sufficient That is, if A z, a0 λ A 0, and the complementary slackness conditions hold, then is the solution to the linear optimization problem Necessity follows directly from Proposition -4 That is, although we interpreted the linear model as an activity model of a firm with n plants, the formal analysis applies to any linear constrained optimization problem Moreover because the problem is linear it is concave Then by Proposition 2-, any satisfying the necessary condition is a solution to the maimization problem That is, the necessary conditions are also sufficient John Riley 4 October 202

6 Essential Microeconomics -6- Non-Linear optimization problems The FOC for non-linear optimization problems are obtained by approimating the non-linear problem by a linear programming problem and then appealing to the Fundamental Theorem of Linear Programming The following simple lemma plays a central role Lemma 4-2: If f and µ sufficiently close to, f( ) > f( ) > ( ) ( ) 0 then for all conve combinations µ of and, To understand this lemma consider the contour set through and the tangent hyperplane For any point to the right of the tangent plane (so that ( ) ( ) > 0 ) consider µ = µ + µ, a conve combination of and ( ) For µ sufficiently small, this must lie in the upper contour set of f since the contour set and tangent plane have the same slope at John Riley 4 October 202

7 Essential Microeconomics -7- Proof of the lemma Define the function µ g( µ ) f( ) = f(( µ ) + µ ) = f( + µ ( )) Note that g(0) = f( ) and g () f( ) = We wish to prove that for µ > 0 and sufficiently small g( µ ) > g(0) To do so we show that g (0) > 0 From the definition of g, g µ = µ ( ) ( ) ( ) Fig 4-: Upper contour sets of f and Setting µ = 0, g = (0) ( ) ( ) By hypothesis this is strictly positive QED John Riley 4 October 202

8 Essential Microeconomics -8- Non-Linear optimization problems As a first step we consider an optimization problem with linear constraints and a nonlinear objective function Because the constraints are linear, the feasible set X is conve P : Ma{ f ( ) X} Suppose that arg Ma{ f ( ) X} We assume that ( ) 0 To establish that the Kuhn Tucker conditions are necessary conditions we show that if solves the problem P, then it must also solve the following linear maimization problem P : Ma{ ( ) X} John Riley 4 October 202

9 Essential Microeconomics -9- Proof by contradiction If does not solve problem P, then there is some X such that >, that is, ( ) ( ) > ( ) ( ) 0 Since X is conve, all conve combinations, µ, of and are in X Also by Lemma 4-2, for all µ sufficiently small, µ lies in the interior of the upper contour set, that is Fig 4-: Upper contour sets of f and µ f( ) > f( ) But then does not solve problem P QED John Riley 4 October 202

10 Essential Microeconomics -0- The general non-linear optimization problem Now consider the general non-linear problem P : Ma{ f ( ) X} where X = { n h( ) 0, i =,, m} + i We will assume throughout that all functions are continuously differentiable Suppose that arg Ma{ f ( ) X} If ( ) = 0, the Kuhn-Tucker conditions (see Proposition 2-2) necessarily hold because we can set the vector of shadow prices equal to zero Henceforth suppose that ( ) 0 Consider also the following linearized problem P : Ma{ ( ) X} n hi where X = { + ( ) ( ) 0, i B} where B is the inde set of nonlinear constraints that are binding at Appealing repeatedly to Lemma 42-, if solves P it must also solve P (See EM section 4) John Riley 4 October 202

11 Essential Microeconomics -- The Kuhn-Tucker conditions are then obtained by applying the Fundamental Theorem of Linear Programming to the linearized problem P Proposition 4-3: Necessary conditions Suppose that solves P : Ma{ f ( ) X} and the following constraint qualifications hold (i) hi int X is non-empty and (ii) ( ) 0, i B, where B is the set of binding constraints Define n hi X = { + ( ) ( ) 0, i B} Then solves P : Ma{ ( ) X} John Riley 4 October 202

12 Essential Microeconomics -2- Appealing to the Fundamental Theorem of Linear Programming we have the following proposition Proposition 4-4: Kuhn-Tucker Conditions Suppose that solves Ma{ f ( ) X} where X { 0, h( ) 0, i =,, m} Let X be the set of linearized binding constraints at If (i) the interior of X is non-empty and (ii) hi for each binding constraint ( ) 0, then the Kuhn-Tucker conditions must hold at i * John Riley 4 October 202

13 Essential Microeconomics -3- Appealing to the Fundamental Theorem of Linear Programming we have the following proposition Proposition 4-4: Kuhn-Tucker Conditions Suppose that solves Ma{ f ( ) X} where X { 0, h( ) 0, i =,, m} Let X be the set of linearized binding constraints at If (i) the interior of X is non-empty and (ii) hi for each binding constraint ( ) 0, then the Kuhn-Tucker conditions must hold at i When are the necessary conditions sufficient? Finally we establish conditions under which the Kuhn-Tucker conditions are sufficient Proposition 4-5: Sufficient condition for a maimum Suppose that the constraint qualifications and Kuhn-Tucker conditions hold at If (i) each of the binding constraints bounds a conve set (ii) the upper contour set { f( ) f( )} is conve and (iii) ( ) 0, then is optimal John Riley 4 October 202

14 Essential Microeconomics -4- Sketch of the proof Consider the two constraint case depicted in Figure 4-3 If satisfies the Kuhn-Tucker conditions and constraint qualifications, then by the Fundamental Theorem of Linear Programming is the solution of the linearized problem P Given the conveity of the constraint functions, the linearized constraint set, X, contains the original set X Fig 4-3: Sufficient conditions for a maimum Given the conveity of the upper contour set of f, it follows that the line tangent to the contour set of f through is a separating line Thus the intersection of the set { f( ) > f( )} and X is empty John Riley 4 October 202

15 Essential Microeconomics -5- If the informal sketch is not convincing, the formal proof appeals to the following mathematical lemma (which is also proved below) Lemma 4-6: If the upper contour set { g ( ) g ( )} is conve, (i) g ( ) g ( ) g ( ) ( ) 0 and (ii) g ( ) > g ( ) g ( ) ( ) > 0 g g and ( ) 0 then John Riley 4 October 202

16 Essential Microeconomics -6- Proof of Proposition 4-5 (Sufficiency) hi By Lemma 4-6, if hi( ) hi( ) = 0 bounds a conve set and ( ) 0, then hi hi( ) hi( ) ( ) ( ) 0 Thus X X ** John Riley 4 October 202

17 Essential Microeconomics -7- Proof of Proposition 4-5 (Sufficiency) hi By Lemma 4-6, if hi( ) hi( ) = 0 bounds a conve set and ( ) 0, then hi hi( ) hi( ) ( ) ( ) 0 Thus X X Suppose that the Kuhn-Tucker conditions hold at Consider the linearized problem at Because the Kuhn-Tucker conditions are both necessary and sufficient for this linear optimization problem, it follows that solves Ma{ ( ) X} Because X X it follows that solves Ma({ ( ) X} * John Riley 4 October 202

18 Essential Microeconomics -8- Proof of Proposition 4-5 (Sufficiency) hi By Lemma 4-6, if hi( ) hi( ) = 0 bounds a conve set and ( ) 0, then hi hi( ) hi( ) ( ) ( ) 0 Thus X X Suppose that the Kuhn-Tucker conditions hold at Consider the linearized problem at Because the Kuhn-Tucker conditions are both necessary and sufficient for this linear optimization problem, it follows that solves Ma{ ( ) X} Because X X it follows that solves Ma({ ( ) X} By Lemma 4-5 (ii), because { f( ) f( )} is conve, f( ) > f( ) ( ) > ( ) ) solves Ma({ f ( ) X} Thus QED John Riley 4 October 202

19 Essential Microeconomics -9- It remains to prove Lemma 4-6 Lemma 4-6: If the upper contour set { g ( ) g ( ) 0} is conve, (i) g ( ) g ( ) g ( ) ( ) 0 and (ii) g ( ) > g ( ) g ( ) ( ) > 0 g g and ( ) 0 then Proof of (i): Consider a vector in the upper contour set Since the upper contour set is conve, every conve combination λ = ( λ) + λ also lies in this set Therefore for any λ (0,), λ dh h( λ ) = g ( ) g ( ) 0 But h (0) = 0, so (0) 0 dλ Since the derivative of h( λ ) is dh g 0 ( ) ( λ λ = ) ( ) dλ it follows that dh g 0 (0) = ( ) ( ) 0 dλ QED John Riley 4 October 202

20 Essential Microeconomics -20- Proof of (ii): From (i) we know that g g ( ) > g ( ) ( ) ( ) 0 Either (ii) holds, or there is some such that g ( ) > g ( ) and g ( ) ( ) = 0 2 g Consider = δ where δ = ε ( ) and ε > 0 Since sufficiently small, 2 g ( ) = g ( δ ) > g ( ) g ( ) > g ( ) It follows that, for ε Also g 2 0 g 2 g ( ) ( ) = ( ) ( ) + ( ) ( ) But the second term on the right-hand side is zero and established that 2 g = ε ( ) We have therefore 2 0 g ( ) > g ( ) and g ( ) ( ) g = ε ( ) g ( ) < 0 But this result contradicts (i) so there can be no such 2 QED John Riley 4 October 202