Pacific Journal of Mathematics


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1 Pacific Journal of Mathematics MINIMIZATION OF FUNCTIONS HAVING LIPSCHITZ CONTINUOUS FIRST PARTIAL DERIVATIVES LARRY ARMIJO Vol. 16, No. 1 November 1966
2 PACIFIC JOURNAL OF MATHEMATICS Vol. 16, No. 1, 1966 MINIMIZATION OF FUNCTIONS HAVING LIPSCHITZ CONTINUOUS FIRST PARTIAL DERIVATIVES LARRY ARMIJO A general convergence theorem for the gradient method is proved under hypotheses which are given below. It is then shown that the usual steepest descent and modified steepest descent algorithms converge under the some hypotheses. The modified steepest descent algorithm allows for the possibility of variable stepsize. For a comparison of our results with results previously obtained, the reader is referred to the discussion at the end of this paper. Principal conditions* Let / be a realvalued function defined and continuous everywhere on E n (real Euclidean wspace) and bounded below E n. For fixed x 0 e E n define S(x 0 ) = {x: f(x) ^ f(x 0 )}. The function^ / satisfies: condition I if there exists a unique point x* e E n such that f(x*) = inf/(a); Condition II at x 0 if fe C 1 on S(x 0 ) and Vf(x) = 0 xee n for x e S(x 0 ) if and only if x = x*; Condition III at x 0 if /e C 1 on S(x 0 ) and Ff is Lipschitz continuous on S(x 0 ), i.e., there exists a Lipschitz constant K> 0 such that Ff(y) Ff(x) \ S K\y x\ for every pair x, yes(x 0 ); Condition IV at x 0 if fec 1 on S(x 0 ) and if r > 0 implies that m(r) > 0 where ra(r) = inf Ff(x), S r (x Q ) = S r Π S(x 0 ), S r = χes r (χ 0 ) {x : I x x* I ^r}, and #* is any point for which f(x*) inf f(x). (If xee n S r (x 0 ) is void, we define m(r) = oo.) It follows immediately from the definitions of Conditions I through IV that Condition IV implies Conditions I and II, and if S(x 0 ) is bounded, then Condition IV is equivalent to Conditions I and II. 2. The convergence theorem* In the convergence theorem and its corollaries, we will assume that / is a realvalued function defined and continuous everywhere on E n, bounded below on E n, and that Conditions III and IV hold at x 0. THEOREM. // 0 < δ g 1/41SΓ, then for any xes(x 0 ), the set (1) S*(a, δ) = {x λ : x λ = x Wf{x), λ > 0, f(x λ )  f(x) ^  δ \Ff(x)\ 2 } Received January 30, The research for this paper was supported in part by General Dynamics/Astronautics, San Diego, California, Rice University, Houston, Texas, and the Martin Company, Denver, Colorado. The author is currently employed by the National Engineering Science Company, Houston, Texas.
3 2 LARRY ARMIJO is a nonempty subset of S(x 0 ) and any sequence {x k }ΐ= 0 such that x k+1, d), k = 0, 1, 2,, converges to the point x* which minimizes f. Proof. If xe S(x 0 ), x λ = x  Wf(x) and 0 ^ λ ^ 1/Z", Condition III and the mean value theorem imply the inequality f(x λ ) f(x) ^  (λ  X 2 K) I Pf(x) 2 which in turn implies that x λ e S*(x, δ) for λ x ^ λ ^ λ 2, λ< so that S*(x, d) is a nonempty subset of S(x Q ). If {^fc )Γ=o is any sequence for which x k+1, δ), k =0,1, 2,, then (1) implies that sequence {f(x k )}~= 0, which is bounded below, is monotone nonincreasing and hence that Vf(x k ) \ * 0 as k > oo. The remainder of the theorem follows from Condition IV. COROLLARY 1. (The Steepest Descent Algorithm) If x k+1 = x k  Lj7/(a? 4 ), k = 0, 1, 2, Δίί. then the sequence {x k }^=Q converges to the point x* which minimizes f. Proof. It follows from the proof of the convergence theorem that the sequence {x k }^= 0 defined in the statement of Corollary 1 is such that x k+1, 1/4Z), k = 0,1, 2,.. COROLLARY 2. (The Modified Steepest Descent Algorithm) If a is an arbitrarily assigned positive number, a m = a/2 m ~\ m = 1, 2,, and x k+1 = x k cc mj Pf(x k ) where m k is the smallest positive integer for which ( 2) f(x k  a m /f(x k ))  f(x k ) ^  \a mh \ Ff(x k ) 2, k = 0, 1, 2,, then the sequence {x k }ΐ= 0 converges to the point x* which minimizes f. Proof. It follows from the proof of the convergence theorem that if x e S(x Q ) and x λ =x  \Pf(x), then f(x λ )  f(x) ^  (1/2) λ Vf(x) 2 for 0 ^ λ ^ 1J2K. If a ^ l/2iγ, then for the sequence {x k }ΐ= 0 in the statement of Corollary 2, m k = 1 and x k+1, (l/2)a), k = 0,1, 2,. If a > 1/2Z", then the integers m^ exist and a mfc > l/4iγ so that
4 MINIMIZATION OF FUNCTIONS 3 3* Discussion* The convergence theorem proves convergence under hypotheses which are more restrictive than those imposed by Curry [1] but less restrictive than those imposed by Goldstein [2]. However, both the algorithms which we have considered would be considerably easier to apply than the algorithm proposed by Curry since his algorithm requires the minimization of a function of one variable at each step. The method of Goldstein requires the assumption that fec 2 on S(x 0 ) and that S(x 0 ) be bounded. It also requires knowledge of a bound for the norm of the Hessian matrix of / on S(x 0 ), but yields an estimate for the ultimate rate of convergence of the gradient method. It should be pointed out that the modified steepest descent algorithm of Corollary 2 allows for the possibility of variable stepsize and does not require knowledge of the value of the Lipschitz constant K. The author is indebted to the referee for his comments and suggestions. REFERENCES 1. H. B. Curry, The method of steepest descent for nonlinear minimization problems, Quart. Appl. Math. 2 (1944), '2. A. A. Goldstein, Cauchy's method of minimization, Numer. Math. 4 (2), (1962),
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6 PACIFIC JOURNAL OF MATHEMATICS H. SAMELSON Stanford University Stanford, California R. M. BLUMENTHAL University of Washington Seattle, Washington EDITORS *J. DUGUNDJI University of Southern California Los Angeles, California RICHARD ARENS University of California Los Angeles, California ASSOCIATE EDITORS E. F. BECKENBACH B. H. NEUMANN F. WOLF K. YOSIDA UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY UNIVERSITY OF SOUTHERN CALIFORNIA SUPPORTING INSTITUTIONS STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON * * * AMERICAN MATHEMATICAL SOCIETY CHEVRON RESEARCH CORPORATION TRW SYSTEMS NAVAL ORDNANCE TEST STATION Printed in Japan by International Academic Printing Co., Ltd., Tokyo Japan
7 Pacific Journal of Mathematics Vol. 16, No. 1 November, 1966 Larry Armijo, Minimization of functions having Lipschitz continuous first partial derivatives Edward Martin Bolger and William Leonard Harkness, Some characterizations of exponentialtype distributions James Russell Brown, Approximation theorems for Markov operators Doyle Otis Cutler, Quasiisomorphism for infinite Abelian pgroups Charles M. Glennie, Some identities valid in special Jordan algebras but not valid in all Jordan algebras Thomas William Hungerford, A description of Mult i (A 1,, A n ) by generators and relations James Henry Jordan, The distribution of cubic and quintic nonresidues Junius Colby Kegley, Convexity with respect to EulerLagrange differential operators Tilla Weinstein, On the determination of conformal imbedding Paul Jacob Koosis, On the spectral analysis of bounded functions JeanPierre Kahane, On the construction of certain bounded continuous functions V. V. Menon, A theorem on partitions of massdistribution Ronald C. Mullin, The enumeration of Hamiltonian polygons in triangular maps Eugene Elliot Robkin and F. A. Valentine, Families of parallels associated with sets Melvin Rosenfeld, Commutative Falgebras A. Seidenberg, Derivations and integral closure S. Verblunsky, On the stability of the set of exponents of a Cauchy exponential series Herbert Walum, Some averages of character sums
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