Hessian sufficiency for bordered Hessian

Res. Lett. Inf. Math. Sci., 2005, Vol. 8, pp 189-196 189Available online at /research/letters/Hessian sufficiency for bordered HessianE RIC I KSOON I MDepartment of Economics, College of Business and Economics,University of Hawaii at Hilo, USAeim@We show that the second–order condition for strict local extrema in both constrained andunconstrained optimization problems can be expressed solely in terms of principal minorsof the (Lagrengean) Hessian. This approach unifies the determinantal tests in the sense thatthe second-order condition can be always given solely in terms of Hessian matrix.1 IntroductionIn the theory of constrained optimization, we use the bordered Hessian determinantalcriterion to test whether an objective function has an extremun at a critical point.However, the required signs of the minors in the case of the constrained optimizationare quite different from those in the case without constraints, which is somewhatconfusing. In this paper, we show that when the constraints are twice differentiable wedo not need the bordered Hessian at all for determinantal test. We only need the(Lagrangian) Hessian matrix for the determinantal test for both unconstrained andconstrained optimization problems. This saves the unnecessary switching from theHessian matrix to the bordered Hessian matrix for determinantal test for the second-order sufficient condition when the optimization problem is subject to constraints..2 DiscussionTo set the stage, first we formally state the standard constrained optimization problemand the second-order sufficient condition, then address the issue of unified signrequirements for the second-order condition for optimization.Let :S φ→\ be a real-valued function defined on a set S in , anda vector function defined on S . Let c be an interior point of S andlet be a point in . Define the Lagrangian function n \:(m g S m n →<\)A m \:S ψ→\ by the equation)()()(x g x x A ′−=φψ, (1)190 E.I. Im where )(x φ and both are twice differentiable at c , the )(x g n m × Jacobian matrixhas a full row rank m . Then, the sufficient first- and second-order conditions for)(c Dg )(x φ to have a strict local minimum (maximum) at c are respectively (Magnus andNeudecker, pp.135-318 for example):0)();(==dx c Dg dx c d ψ and for all , (2)0)(=c g n dx ℜ∈0)();(2>′=dx c H x d dx c d ψψ for all 0≠dx satisfying 0);(=dx c dg . (3)where , and D and H are respective notations for Jacobian andHessian operators.∑=−≡mm i i i c Hg H H )(λφψThe components of the second-order condition in (3) can be consolidated into one asfollows. Partitioning dx and conformably as )(c Dg B n m ≡×)(′′′=−m n m x d x d dx # and, we can rewrite )()(,m n m m B B c g D −=#0)(=c dg in the second order condition:0)(),(,=+==−−m n m n m m m dx B dx B dx c Dg dx c dgfrom which .(4) m n m n m m m dx B B dx −−−−=,1Incorporating (4) into (3), two components of the second-order condition are nowconsolidated into one:)0(0),(2<>′′=−−m n m n dx Q H Q x d dx c d ψψ for 0≠dxwhere , or, equivalently )(1,′′−=−−−m n m m n mI B B Q #)0(0)()(<>′≡ΩQ c H Q c ψ. (5)The strict inequality holds if and only if the signs of the principal minors ofadhere to:)(c Ω0>Ωk for strict local minima;0)1(>Ω−k k for strict maxima ( k = m+1, m +2, …, n -m ) (6)where ; in which kk k E c E ′Ω=Ω)(())(0k n k k k I E −×=# and k k E Q Q ′=.In practice, however, we test for the sign definiteness of the second-order condition,using the determinantal criterion based on the bordered Hessian matrix:0)()1(>−c H r m ψ for strict local minima;0)()1(>−c H r r ψ for strict local maxima, ),...,1(n m r += (7)where: ()⎥⎦⎤⎢⎣⎡′=×××r r m r m m m r c H B B c H )(0)(ψψ (8)Hessian sufficiency for bordered Hessian191defines the principal submatrices of bordered Hessian of order r m +of the borderedHessian matrix of order :n m +⎥⎦⎤⎢⎣⎡′=×××)(0)(c H B B c H n m n m m m ψψ (9) in which denotes the first r columns of r m B ×)(c g D B n m ≡×However, the sufficient second-order condition for a strict local optimum can bestated purely in terms of principal minors of )(c H ψ instead of those of the borderedHessian as discussed in the following section.3 Hessian Sufficiency for Bordered HessianIn the Hessian alternative to the bordered-Hessian, it is essential to note that there is arank condition implicit in the first-order condition, which is not needed in the borderedHessian approach. To make the point, re-express the first part of first-order condition in(2) in variable form for further differentiation:. for all n m n m dx −−∈\0)();(==dx x D dx x d ψψDifferential of this first-order condition at c x =, with (4) incorporated, can be writtenas:dx c H x d dx c d )(),(2ψψ′=0)(=Ω′=−−m n m ndx c x d for all (10)m n m n dx −−ℜ∈†Though Equation (10) contains the Hessian matrix, it is not the second-order condition.It is in essence the first-order condition in quadratic form which facilitates identifying auseful property of Q c H Q )(ψ′ associated with a strict local critical point.Suppose that . Then, 0n m dx −≠0)(=Ω−m n dx c for all (11)0≠−m n dxSince 0≠−m n dx , Ω must be singular, hence )(c Ω=Ω (therefore c ) depends onwhich violates the assumption that c is a strict local critical point. Therefore, shouldbe nonsingular, which in turn implies m n dx −Ω0=−m n dx . Hence, a lemma can be stated withoutproof:LEMMA : Let c be a strict local critical point for the optimization problem (1). Then,Q c H Q c )()(ψ′≡Ωis nonsingular.H † Note that if is linear, )(x g ψ does not depend on x , therefore c . This is exactly the reason whyis assumed to be twice differentiable. However, a linear constraint is technically twicedifferentiable when squared. Hence, our approach applies to both linear and nonlinear constraints.)(x g192 E.I. Im Having established formally that is nonsingular if c is a strict local critical point,we are now ready to restate the second-order sufficient condition in (5) solely in term ofthe (Lagrangian) Hessian instead of bordered Hessian as follows.)(c ΩTHEOREM (General Sufficient Condition): If )(c H ψ is nonnegative definite(nonpositive definite) where c is a strict critical point x , )(c ψ is a strict minimum(maximum), and m n c H Rank −≥))((ψProof : Since is a strict critical point, c Q c H Q c )()(ψ′=Ω is nonsingular (thereforeof full rank) in view of the Lemma. If )(c H ψ is nonnegative definite, it has a full-rankfactorization: T T c H ′=)(ψ where T is of full column rank. Hence, we can rewritewhich is nonnegative definite. Since )()()(Q T Q T c ′′′=Ω)(c Ω is both non-singular andnonnegative definite, which implies that 0)(>Ωc )(c ψ is a strict local maximum. Theproof for a strict local maximum is similar. The second part of the theorem can beproved as follows. implies 0)(>Ωc m n c H Rank −≥))((ψ: Since, m n c Rank −=Ω))((m n Q T Rank −=′)(. However, since byassumption, m n Q Rank −=)(m n T Rank −≥)( from which follows m n c H Rank −≥))((ψ. )QEDCOROLLARY (Special Sufficient Condition): If )(c H ψ is positive definite(negative definite) where c is a strict local critical point, )(c ψis a strict minimum(maximum), and n c (H Rank =))(ψ.‡Proof : Since positive definiteness (negative definiteness) is a special case ofnonnegativity (nonpositivity), the corollary follows from the theorem as a trivial case.Since definite matrices are of full rank, n c H Rank =))((ψ. QED4 An ExampleSolve the problem (Magnus & Neudecker, pp. 138-139)22),()()(w v x Min Max w v x +==φsubject to .322=++w vw vThen, the Lagrangian function will be: ,)3()(2222−++−+=w vw v w v x A ψ ‡The special sufficient condition has been known and noted in a number of places (e.g., Hal R. Varian(p. 498)), but not the general sufficient condition.Hessian sufficiency for bordered Hessian193and the first-order conditions are:3;022;02222=++=−−=−−w vw v w v w w v v A A A A .Solving the first-order conditions for control variable vector ),(w v x =, we have threestrict local stationary points: 321,,c c c x = are defined respectively as)1,1(),(111−−==w v c , )3,3(),(222−==w v c , and )3,3(),(333−==w v c .The corresponding Lagrangian multipliers are, respectively, 3/21=A , , and. The (Lagrangian) Hessian and bordered Hessian are, respectively:22=A 23=A.222222220)(;2222)(⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+++=⎥⎦⎤⎢⎣⎡−−−−=λλλλψλλλλψw v w v w v w v x H x H Now, we check the sign behavior of all minors, not just the leading minors, of )(x H ψat each of the three critical points to see whether we have a local minimum or maximumor neither in view of the Lemma.Minors of )(1c H ψ for the first stationary point are: 0~;03/2)3/2(22~21=>=−=M M .)(1c H ψ is positive semi-definite of rank 1, hence, in view of the Theorem, the firstcritical point is a strict local minimum.Corresponding principal minors of )(2c H ψ and )(3c H ψ are identical:0~;02~21=<−=M M .Both )(2c H ψ and )(3c H ψ are negative semi-definite of rank 1, thus in light of theTheorem )(x φ has two strict local maxima.(The same conclusion is arrived for each critical point by way of the bordereddeterminantal criterion in Magnus and Neudecker).5 Linear Constraints as Twice DifferentiablesTypical linear constraints can be transformed into twice differentiables by squaring thelinear constraints. Therefore, the Hessian alternative to bordered Hessian applies to all194 E.I. Im constraints in general. This can be illustrated by a simple example in Chiang andWainwright (Example 2, p. 360):Find the extremum ofvw z = subject to 6=+w v . (I)The Hessian and bordered Hessian matrices are respectively:.011101110)(;0110)(⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎦⎤⎢⎣⎡=x H x H ψψThe determinant of bordered Hessian is positive: 02)(>=x H ψ which meets the signrequirement for a strict local maximum, whereas the leading principal minors of Hessian are: 01)(2;0)(1<−==x H x H ψψ which do not meet the sign requirements for a local maximum. Clearly, the Hessian approach fails to identify the localmaximum. The reason is that theorem is applicable only when the constraint is twicedifferentiable.This problem can reformulated using the squared constraint asvw z = subject to or . (II)226)(=+w v 36222=++w vw vThen, the Lagrangian function for (II) will be: , andthe first-order conditions are:)362()(22−++−=w vw v vw x A ψ362;022;02222=++=−−=−−w vw v w v v w v w A A A A .The Lagrangian Hessian then will be:.221)(2212)(2)(2)(20)(;221212)(⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−−+−+−+−=⎥⎦⎤⎢⎣⎡−−−−=A A A A A A A A w v w v w v w v x H x H ψψSolving the first-order conditions for control vector ),(w v x =, we have two strict localcritical points:)3,3(),(1==w v c and )3,3(),(2−−==w v c and 4/1=A for both critical points.Hessian sufficiency for bordered Hessian195However, since does not satisfy the original constraint, it will be out ofconsideration. At 2c )3,3(),(1==w v c ,⎥⎦⎤⎢⎣⎡−−=⎥⎦⎤⎢⎣⎡−−−−=2/12/12/12/1221212)(1A A A A c H ψ for which: 0~;02/1~21=<−=M M)(c H ψ is negative semi-definite of rank 1, hence, in view of the Theorem, the firstcritical point is a strict local maximum, which is exactly the same results as would beproduced if we used the bordered Hessian determinantal rule to Problem (II).The Hessian Approach to Problem I breaks down for the simple reason that theconstraint is not twice differentiable as required by the Theorem, but it works forProblem II since the constraint are converted into a twice differentiable.6 ConclusionIn this paper, we have shown that the conventional second-order sufficient condition forconstrained optimization in terms of minors of bordered Hessian can be recastalternatively in terms of minors of Hessian matrix, making it possible to represent thesecond-order condition in general all in terms of minors of the Hessian matrix. Thistheoretical result dispels the misconception that borders in the bordered Hessian matrixhave some bearings on the second-order condition. Furthermore, the second-ordercondition based on Hessian matrix is more convenient in practice than the borderedHessian since finding minors of Hessian is less cumbersome than those of borderedHessian. Practitioners of the determinantal tests in optimization theory find it a bitconfusing to use the minors of Hessian for constrained optimization problems butminors of bordered Hessian for unconstrained problems. Finally, the theorem seeminglyapplicable only to the optimization problems with twice differentiable constraintsindeed applies to problems with linear constraints, therefore applicable in general.APPENDIXAs shown in Magnus and Neudecker ( p.55),k k m r m T x H Ω−=−2)1()()1(ψ )...,,2,1;(m n k k m r −=+= (A1)where (nonsingular) is defined as: . (A2) k T ⎟⎟⎠⎞⎜⎜⎝⎛=×××k mk k m m m k I B B T 0196 E.I. Im From (A1) for readily follows: m n k −=m n m n n m T x H −−Ω=−2)()1(ψ. (A.3)It is clear from (A.3) that 0≠Ωif and only if 0)(≠x H ψ, or equivalently Ω is nonsingular if and only if )(x H ψ is nonsingular. Note that )(x H H n ψψ= andwhich is obvious from the context.m n −Ω=ΩReferencesChiang A. and K. Wainwright (2005). Fundamental Methods of MathematicalEconmics . Fourth Edition. McGraw-Hill, New York.Magnus, J. R. and H. Neudecker (1998). Matrix Differential Calculus withApplications in Statistics and Econometrics . Revised Edition. Wiley, New York.Varian, H. R. (2001). Microeconomic Analysis . Norton & Company, New York.。