In this note we show that the strong duality theorem of an unconstrained (generalized) geometric
programming problem as defined by Peterson (cf.[1]) is actually a special case of a Lagrangian
duality result. Contrary to [1] we also consider the case that the set C is compact and convex
and in this case we do not need to assume the standard regularity condition

Frenk, J.B.G. (Hans)

Still, G.J.

Frenk, JBG (Hans)

Still, G

EUR Research Repository

A note on the dual of an unconstrained (generalized) geometric programming problem

In this note we show that the strong duality theorem of an unconstrained (generalized) geometric\nprogramming problem as defined by Peterson (cf.[1]) is actually a special case of a Lagrangian\nduality result. Contrary to [1] we also consider the case that the set C is compact and convex\nand in this case we do not need to assume the standard regularity condition

Erasmus University Digital Repository

          ERIM REPORT SERIES RESEARCH IN MANAGEMENT ERIM Report Series reference number ERS-2005-006-LIS (Logistics) Publication  February 2005 Number of pages 7 Email address corresponding author frenk@few.eur.nl Address Erasmus Research Institute of Management (ERIM) Rotterdam School of Management / Rotterdam School of Economics  Erasmus Universiteit Rotterdam P.O. Box 1738  3000 DR Rotterdam, The Netherlands Phone:  +31 10 408 1182  Fax: +31 10 408 9640 Email:  info@erim.eur.nl Internet:  www.erim.eur.nl  Bibliographic data and classifications of all the ERIM reports are also available on the ERIM website:  www.erim.eur.nl A NOTE ON THE DUAL OF AN UNCONSTRAINED (GENERALIZED) GEOMETRIC PROGRAMMING PROBLEM  J.B.G. Frenk and G.J. Still ERASMUS  RESEARCH  INSTITUTE  OF  MANAGEMENT  REPORT SERIES RESEARCH IN MANAGEMENT   BIBLIOGRAPHIC DATA AND CLASSIFICATIONS Abstract In this note we show that the strong duality theorem of an unconstrained (generalized) geometric programming problem as defined by Peterson (cf.[1]) is actually a special case of a Lagrangian duality result. Contrary to [1] we also consider the case that the set C is compact and convex and in this case we do not need to assume the standard regularity condition. Mission:  HF 5001-6182  Programme: TX301-329 Library of Congress Classification  (LCC)    LCC Webpage   Paper: HB 143 Mathematical programming Mission: M  Programme : M 11, R 4 Journal of Economic Literature  (JEL)  JEL Webpage   Paper: C 69 Mathematical Methods and Programming, Other. Gemeenschappelijke Onderwerpsontsluiting (GOO) Mission: 85.00  Programme: 85.34 Classification GOO   Paper: 31.80 Toepassingen van de wiskunde Mission: Bedrijfskunde / Bedrijfseconomie  Programme: logistiek, management Keywords GOO   Paper: geometrische kwantisatie, lagrange functies Free keywords Generalized geometric programming, Lagrangian dual, regularity conditions       A note on the dual of an unconstrained(generalized) geometric programming problem.J.B.G.Frenk∗G.J.Still†February 13, 2005AbstractIn this note we show that the strong duality theorem of an uncon-strained (generalized) geometric programming problem as defined byPeterson (cf.[1]) is actually a special case of a Lagrangian duality result.Contrary to [1] we also consider the case that the set C is compact andconvex and in this case we do not need to assume the standard regularitycondition.Keywords: Generalized geometric programming, Lagrangian dual, regularityconditions.1 Introduction.In [1] a dual problem is introduced for a so-called unconstrained (generalized)geometric programming problem. To introduce this optimization problem wewill use the same notation as in [1]. Although this notation is not standardfor a (convex) cone in convex analysis it is done to facilitate comparing theresult in this note and Theorem 1 in [1]. Let X ⊆ Rn be a nonempty propercone, C ⊆ Rn a nonempty set and g : C → R a real-valued function. Anunconstrained (generalized) geometric programming problem is denoted in [1]by problem A and given byϕ := infx∈X∩C g(x). (A)∗Corresponding author J.B.G.Frenk, Econometric Institute, Erasmus University, PO Box1738, 3000DR Rotterdam, The Netherlands, E-mail: frenk@few.eur.nl†G.J.Still, Department of Mathematical Sciences, University of Twente, P.O.Box 217,7500AE Enschede, The Netherlands, E-mail: g.still@math.utwente.nl1It is argued in [1] that the importance of the above formulation with a propercone X ⊆ Rn is due to the possibility to reformulate a nonseparable opti-mization problem into an instance of problem A with a separable objectivefunction g and a ”separable” proper cone. This procedure is then carried outfor a quadratic optimization problem with linear constraints. Introducing as in[1] the function h : Rn → (−∞,∞] given byh(y) := sup{y>x− g(x) : x ∈ C}. (1)and the setD := {y ∈ Rn : h(y) <∞}the dual problem of problem A is denoted by problem B and this problem hasthe formψ := inf{h(y) : y ∈ Y ∩D} (B)with Y the so-called dual cone of X given byY = {y ∈ Rn : y>x ≥ 0 for every x ∈ X}.In Theorem 1 of [1] a strong duality result between problem A and problem B(under some regularity condition) is given for the case that C is a closed con-vex set, X a closed convex cone and g a proper lower semicontinuous convexfunction on C. In this note we will show that a similar result holds without anyregularity condition if additionally C is compact and g : U → R a real-valuedconvex function on an open set U containing C. To verify this we first give aLagrangian dual interpretation of problem B. In our opinion the Lagrangiandual approach in combination with the natural concept of a relaxation of aproblem is a pedagogically good approach teaching duality theory to studentsin Operations Research and Engineering. At the same time our proof can alsobe easily adapted to the noncompact case as considered in [1]. To do this wefirst let problem A resemble a standard nonlinear programming problem (seefor example [2]) by introducing the binary relation ≤Xgiven by x1 ≤X x2 ifand only if x2 − x1 ∈ X. Observe this binary relation is transitive if X is aconvex cone. Now problem A is the same asinf{g(x) : x ∈ C, x ≥X 0}. (A)Inserting now the restriction x ≥X 0 into the objective function and penalizingit with a Lagrangian dual variable y ∈ Y we introduce the Lagrangian functionθ : Y → (−∞,∞) given byθ(y) := inf{g(x)− y>x : x ∈ C}.2It follows immediately for every y ∈ Y thatθ(y) ≤ inf{g(x)− y>x : x ∈ C, x ≥X 0} ≤ ϕ. (3)Since we like to achieve the valueϕ it is now obvious by relation (3) to considerthe Lagrangian problemϑ := supy∈Y θ(y)By relation (1) it follows that supy∈Y θ(y) = − infy∈Y h(y) and this is thedual problem considered in [1]. Moreover, it follows by (3) that−ψ = ϑ ≤ ϕ.We will now show the following extension of Theorem 1 in [1] forC a compactconvex set. Observe the proof of this result is an adaptation of the proof insubsection 10.1.1 in [2] given for the Lagrangian dual of optimization probleminf{g(x) : x ∈ C, x ≥ 0}. A similar result also appeared on page 150 of[7]. To keep the paper selfcontained we first mention part 2 of Theorem 1 ofPeterson (cf.[1]). Actually the formulation given below is a slight extension ofpart 2 of Theorem 1 in [1]. In the next formulation we do not assume contraryto [1] that the function g : C → R is a lower semicontinuous function.Theorem 1 If X is a closed convex cone, C a convex set, ri(C) ∩ ri(X) isnonempty and g : C → R is a real-valued convex function onC, then it followsthat problem B has an optimal solution and ϕ = −ψ.The condition ri(C) ∩ ri(X) 6= ∅ is a so-called regularity condition andthis condition trivially implies that the feasible setF := {x ∈ C : x ≥X 0}of problem A is nonempty. In this note we will show in the next proof if addi-tionally C is a compact convex set and of course F is nonempty, then the sameresult holds for any real-valued convex function g : U → R with U an open setcontaining X. We also observe that it is easy to adapt the next proof to verifyTheorem 1 (see Remark 2).Proof. We already showed ϑ ≤ ϕ. To prove that ϑ = ϕ we assume by contra-diction that ϑ < ϕ and so there exists some v ∈ R satisfyingϑ < v < ϕ. (4)3Introduce now the vector valued function h : C → Rn+1 given by h(x) :=(x, g(x))>. If h(C) := {h(x) : x ∈ C} denotes the range of the function hand R+ = [0,∞) we first observe thath(C) + ((−X)× R+) = {(x− k, r) : r ≥ g(x), x ∈ C, k ∈ X} (5)This shows by the convexity of the function g on U and X a convex cone thatthe set h(C) + ((−X) × R+) is convex. Moreover, since g : U → R is areal-valued convex function on the open set U the function g is continuous onU (cf.[2]). By the compactness of C ⊆ U this implies that the set h(C) iscompact and using X is a closed convex cone we obtain that the set h(C) +((−X) × R+) is also closed. It is now easy to verify by contradiction thatthe vector (0, v) does not belong to h(C) + ((−X) × R+) and so we canstrictly separate the closed convex set h(C) + ((−X) × R+) and the point(0, v) (see for example [6], [4] or [5]). Hence there exists some nonzero vector(y∗, α) ∈ Rn+1 such thatαv < −y>∗ (x− k) + α(g(x) + β) (6)for every x ∈ C, β ≥ 0 and k ∈ X. Since β ≥ 0 and k ∈ X in relation(6) it must follow by contradiction that α ≥ 0 and y∗ ∈ Y. If we assume bycontradiction that α = 0 we obtain by relation (6) and 0 ∈ −X (−X is aclosed convex cone!) that y∗ ∈ Y is nonzero andy>∗ x < 0 (7)for every x ∈ C. Since by assumption the feasible region F = {x ∈ C :x ≥X 0} is nonempty and y∗ ∈ Y one can find some x0 ∈ F ⊆ C satisfyingy>∗ x ≥ 0 and this contradict relation (7). Hence α > 0 and by relation (6) weobtainv < g(x)− α−1y>∗ xfor every x ∈ C. Since α−1y∗ ∈ Y this implies by relation (4) that v <θ(α−1y∗) ≤ ϑ < v and we have a contradiction. This proves the desiredresult. Although we did not give a proof of Theorem 1 it is easy to adapt theabove proof to show this result. How to change this proof is discussed in thenext remark.Remark 2 If the convex setC is not compact it still follows that the setH(C)+(R+ × (−X)) is convex for g : C → R a real-valued convex function on theconvex set C and (0, v) does not belong to H(C) + ((−X)× R+). However,4the set H(C)+ ((−X)×R+) might not be closed and so we cannot apply thestrict separation result. However, we can apply in this case a slight extensionof the strict separation result given by the proper separation result between aconvex set and a point outside this set (see for example Theorem 1.2 of [4] or[6]). Since in the proper separation result it follows that (y∗, α) is nonzero andy∗ can be selected from the affine hull of the set C −X and it is well-known([6]) that ri(C −X) = ri(C)− ri(X) the above proof can be easily adaptedto verify Theorem 1. Actually one can use this proof to show a strong dualityresult for the Lagrangian dual of a more general optimization problem (cf.[3]).To extend Theorem 1 to nonconvex functions g and C not convex we firstintroduce the m-dimensional unit simplex∆m := {λ ∈ Rm : λi ≥ 0,∑mi=1λi = 1}.It follows for every x belonging to conv(C) that there exists by definition some{x1, ..., xm} ⊆ C,m ∈ N satisfyingx =∑mi=1λixi, λ ∈ ∆m. (8)By the definition of the Lagrangian function θ and relation (8) this yieldsθ(y) ≤ inf1≤i≤m{g(xi)− y>xi} ≤∑mi=1λig(xi)− y>x (9)for every y ∈ Y. If the function g : C → R is bounded below by a affinefunction on C it is shown in Proposition 2.5.1 of [5] that the biggest convexfunction co(g) : conv(C)→ R majorized by g on C is given byco(g)(x) := inf{∑mi=1λig(xi) :x =∑mi=1 λixiλ ∈ ∆m, xi ∈ C ,m ∈ N}(10)This implies by relation (9) thatθ(y) ≤ co(g)(x)− y>xfor every x ∈ conv(C) and y ∈ Y. Hence by the definition of the Lagrangianfunction θ and co(g) we obtainθ(y) = inf{co(g)(x)− y>x : x ∈ conv(C)} (11)for every y ∈ Y and using Theorem 1 the next result follows.5Theorem 3 If X is a closed convex cone, ri(conv(C)) ∩ ri(X) is nonemptyand g : C → R is bounded below by an affine function on C, then it followsthat problem B has an optimal solution andsupy∈Y θ(y) = inf{co(g)(x) : x ∈ conv(C), x ≥X 0}.If the set C ⊆ Rn is compact we observe by Lemma 1.8 of [4] that theset conv(C) is also compact. To formulate now Theorem 3 for C compactwe assume additionally that the real-valued function g is defined on an openbounded set U containing C and that g : U → R is bounded below (on thisopen set) by an affine function. This implies by relation (10) that the convexfunction co(g) : conv(U) → R is bounded below by an affine function onthe open convex set conv(U) ⊆ conv(C). Hence the convex function co(g)is real-valued on conv(U) and so it is continuous on conv(U) ⊆ conv(C).Replacing now C by conv(C) and g by co(g) in the previous proof and ap-plying relation (11) it follows that Theorem 3 also holds for C compact, thereal-valued function g : U → R is bounded below by an affine function on anopen bounded set U containing C and conv(C) ∩X is nonempty.References[1] E.L.Peterson. The fundamental relation between geometric programmingduality, parametric programming duality and ordinary Lagrangian duality.Annals of Operations Research, 105:109–153, 2001.[2] Faigle, U., Kern, W and G. Still. Algorithmic Principles of Mathemati-cal Programming, volume 24 of Kluwer Texts in Mathematical Sciences.Kluwer Academic Publishers, Dordrecht, 2002.[3] Frenk, J.B.G and G. Kassay. On classes of generalized convex functions,Gordan-Farkas type theorems and Lagrangian duality. Journal Of Opti-mization Theory and Applications, 102(2):315–343, 1999.[4] Frenk, J.B.G and G. Kassay. Introduction to Convex and QuasiconvexAnalysis. In Hadjisavvas, N., Komlo´si, S and S. Schaible, editor, Hand-book of Generalized Convexity and Generalized Monotonicity, volume 76of Nonconvex Optimization and its Applications, pages 3–87. SpringerVerlag, Dordrecht, 2004.[5] Hiriart-Urruty, J.B and C. Lemare´chal. Convex analysis and minimiza-tion algorithms I, volume 305 of Grundlehren der mathematische Wis-senschaften. Springer Verlag, Berlin, 1993.6[6] Rockafellar, R.T. Convex Analysis, volume 28 of Princeton MathematicalSeries. Princeton University Press, Princeton, 1972.[7] Shapiro, J.F. Mathematical Programming. Wiley, New York, 1973.7Publications in the Report Series Research∗ in Management  ERIM Research Program: “Business Processes, Logistics and Information Systems”  2005  On The Design Of Artificial Stock Markets Katalin Boer, Arie De Bruin And Uzay Kaymak ERS-2005-001-LIS http://hdl.handle.net/1765/1882  A note on the paper Fractional Programming with convex quadratic forms and functions by H.P.Benson J.B.G.Frenk ERS-2005-004-LIS  A note on the dual of an unconstrained (generalized) geometric programming problem J.B.G.Frenk and G.J.Still ERS-2005-006-LIS                                                  ∗  A complete overview of the ERIM Report Series Research in Management: https://ep.eur.nl/handle/1765/1   ERIM Research Programs:  LIS Business Processes, Logistics and Information Systems  ORG Organizing for Performance  MKT Marketing  F&A Finance and Accounting  STR Strategy and Entrepreneurship  

A Note on the Dual of an Unconstrained (Generalized) Geometric Programming Problem

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