VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS NGUYEN THAI AN VARIATIONAL ANALYSIS AND SOME SPECIAL OPTIMIZATION PROBLEMS Speciality: Applied Mathematics Speciality code: 62 46 01 12 DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2016 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS Nguyen Thai An VARIATIONAL ANALYSIS AND SOME SPECIAL OPTIMIZATION PROBLEMS Speciality: Applied Mathematics Speciality code: 62 46 01 12 DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisors: Prof Dr Hab Nguyen Dong Yen Assoc Prof Nguyen Mau Nam HANOI - 2016 Trang (i) i Abstract This dissertation uses tools from variational analysis and optimization theory to study some complex facility location problems involving distances to sets In contrast to the existing facility location models where the locations are of negligible sizes, represented by points, the new approach allows us to deal with facility location problems where the locations are of non-negligible sizes, now represented by sets Our efforts focus not only on studying theoretical aspects but also on developing effective algorithms for solving these problems Besides, we also introduce an algorithm for minimizing the difference of functions Our main results include: - Algorithms based on Nesterov’s smoothing technique and the majorizationminimization principle for solving new models of the Fermat-Torricelli problem - Theoretical properties as well as an algorithm based on the log-exponential smoothing technique and Nesterov’s accelerated gradient method for the smallest intersecting ball problem - Solution existence together with an algorithm based on the DC algorithm and the Weiszfeld algorithm for a nonconvex facility location problem - Convergence analysis of a generalized proximal point algorithm for minimizing the difference of a nonconvex function and a convex function ii Confirmation This dissertation was written on the basis of my research works carried out at the Institute of Mathematics, Vietnam Academy of Science and Technology, under the guidance of Prof Nguyen Dong Yen and Assoc Prof Nguyen Mau Nam All results presented in this dissertation have never been published by others Hanoi, August 2016 The author Nguyen Thai An iii Acknowledgment I would like to express my sincere gratitude to Prof Nguyen Dong Yen and Assoc Prof Nguyen Mau Nam for their guidance and supports I thank them for always being there for me and providing many relevant suggestions and worthy opinions through all stages of this dissertation I am grateful for being able to participate in their research groups where I have had the pleasure of working with many active and accomplished researchers I would like to thank the Board of Directors and the research staff of the Institute of Mathematics, Vietnam Academy of Science and Technology, for providing me with a wonderful scientific environment I am also grateful to Prof Hoang Xuan Phu, Assoc Prof Ta Duy Phuong, Assoc Prof Phan Thanh An and all members of the Weekly Seminar at the Department of Numerical Analysis and Scientific Computing, Institute of Mathematics, for their valuable discussions Financial supports from the Vietnam National Foundation for Science and Technology Development (NAFOSTED), the Vietnam Institute for Advanced Study in Mathematics (VIASM), and Thua Thien Hue College of Education, are gratefully acknowledged My deepest gratitude goes to my parents, my sisters and brothers, for their supports and continuing encouragement I want to thank my loving wife who always believes me in pursuing my dreams I want to thank her for her sacrifices and supports during the past three years Finally, I would like to thank and dedicate this dissertation to my little daughter, Bao Nguyen, who is the greatest inspiration of my life iv Contents Table of Notations vii List of Figures viii Introduction ix Chapter Preliminaries 1.1 Tools of Convex Analysis 1.2 Majorization-Minimization Principle 1.3 Nesterov’s Accelerated Gradient Method 1.4 Nesterov’s Smoothing Technique 1.5 DC Programming and DC Algorithm 1.6 Conclusions 1 10 Chapter Effective Algorithms for Solving Generalized FermatTorricelli Problems 11 2.1 Generalized Fermat-Torricelli Problems 11 2.2 Nesterov’s Smoothing Technique and a General Form of the Majorization-Minimization Principle 13 2.3 Problems Involving Points 17 2.4 Problems Involving Sets 21 2.5 Numerical Examples 32 2.6 Conclusions 34 Chapter The Smallest Intersecting Ball Problem 36 3.1 Problem Formulation and Theoretical Aspects 36 3.2 A Smoothing Technique for the Smallest Intersecting Ball Problem 47 v 3.3 3.4 3.5 A Majorization-Minimization Algorithm for the tersecting Ball Problem Numerical Implementation Conclusions Smallest In 53 59 62 Chapter A Nonconvex Location Problem Involving Sets 64 4.1 Problem Formulation 64 4.2 Solution Existence in the General Case 66 4.3 Solution Existence in a Special Case 73 4.4 A Combination of DCA and Generalized Weiszfeld Algorithm 79 4.5 Conclusions 85 Chapter Convergence Analysis of a Proximal Point Algorithm for Minimizing a Difference of Functions 87 5.1 The Kurdyka-Lojasiewicz Property 87 5.2 A Generalized Proximal Point Algorithm for Minimizing a Difference of Functions 91 5.3 Examples 102 5.4 Conclusions 106 General Conclusions 107 List of Author’s Related Papers 108 References 109 Appendix A 117 Index 127 vi Table of Notations IN := {0, 1, 2, } ∅ IR IRn (a, b) [a, b] median{α, β, γ} x, y |x| x I A bd Ω co Ω cone Ω d(x; Ω) or dist(x; Ω) P (x; Ω) N (x; Ω) {xk } xk → x liminf αk set of natural numbers empty set set of real numbers n-dimensional Euclidean vector space set of x ∈ IR with a < x < b set of x ∈ IR with a ≤ x ≤ b the middle number of the list obtained after sorting α, β, γ from the smallest to the largest canonical inner product absolute value of x ∈ IR Euclidean norm of a vector x n × n unit matrix transposed matrix of a matrix A topological boundary of Ω convex hull of Ω cone generated by Ω distance from x to Ω Euclidean projection from x onto Ω normal cone to Ω at x ∈ Ω sequence of vectors xk converges to x in norm topology lower limit of a sequence {αk } ⊂ IR limsup αk upper limit of a sequence {αk } ⊂ IR k→+∞ k→+∞ δ(·; Ω) f : IRn → IR ∪ {+∞} domf f∗ ∂f (x) ∂ F f (x) ∂ L f (x) indicator function of Ω extended-real-valued function effective domain of f Fenchel conjugate function of f subdifferential of f at x in the sense of convex analysis Fr´echet subdifferential of f at x limiting subdifferential of f at x vii List of Figures 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 4.1 4.2 MM algorithm for a generalized Fermat-Torricelli problem Generalized Fermat-Torricelli problems with different norms A generalized Fermat-Torricelli problem with US Cities A generalized Fermat-Torricelli problem with MM method A smallest intersecting ball problem for 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proj2box(x,a(j,:),r(j)); end for k=1:step2 x = Weiszfeld(x, A); x = proj2line(x, alpha, beta); end v = val_square_EU(x, a, r); val_wei(i) = v; end 117 v_ref = v; disp(’ -WEISZFELD RESULTS -’) toc fprintf(’\nOptimal Solution: [%.2f %.2f] \n’, x(1), x(2)); fprintf(’Optimal Value: %.2f \n\n’, v); %%%% -Stochastic -clear toc tic x = [0 180]; val_sto = 0; step = 10000; % step = to stop Stochastic for k = 1:step temp = randperm(N); index = temp(1); w_bar = proj2box(x, a(index, :), r(index)); if(norm(x - w_bar) ~= 0) x = x - N/k*(x - w_bar)/norm(x - w_bar); x = proj2line(x, alpha, beta); end v = val_square_EU(x, a, r); val_sto(k) = v; if(norm(v-v_ref) < 1e-6) k break; end end disp(’ -STOCHASTIC RESULTS -’) toc fprintf(’\nOptimal Solution: [%.2f %.2f] \n’, x(1), x(2)); fprintf(’Optimal Value: %.2f \n\n’, v); clear toc index = 1; b = [0 0]; for i = 1:N if(mod(i, 3) == 0) b(index, :) = a(i, :); index = index + 1; end end rb = r(1:length(b)); tic figure(2) 118 USmapDrawSquareEU_line(x, a, r, alpha, beta) axis([-180 -65 15 75]); xlabel(’$\mathrm{Longitude}$’, ’Interpreter’,’latex’, ’fontsize’,13); ylabel(’$\mathrm{Latitude}$’, ’Interpreter’,’latex’, ’fontsize’,13); toc % -function USmapDrawSquareEU_line(x, a, ra, alpha, beta) for p=1:length(a) % plot points a plot(a(p, 1), a(p, 2), ’blu+’, ’markersize’, 6) hold on; end x1 = -170; x2 = (1 - alpha*x1)/beta; x3 = -80; x4 = (1 - alpha*x3)/beta; plot([x1 x3], [x2 x4], ’bla’, ’LineWidth’, 2) plot(x(1), x(2), ’r.’, ’markersize’, 20); end % -function v = val_square_EU(x, a, r) v = 0; N = size(a, 1); for m = 1:N w = proj2box(x, a(m, :), r(m)); v = v + norm(x - w); end end % -function f = Weiszfeld(x, a) y = 0; z = 0; count = 0; % check if any equal x for i = 1:length(a) temp = norm(x - a(i,:)); % if x = if(temp == 0) count = 1; break; else y = y + 1/temp; z = z + a(i,:)/temp; end end if(count == 1) f = x; else f = z/y; end end % -function y=proj2box(x,w,r) y=max(w-r, min(x,w+r)); end 119 % -function sol = proj2line(x, alpha, beta) %%% Line: alpha*x(1) + beta*x(2) = a = x(1); b = x(2); s1 = 0; s2 = 0; if(alpha == 0) %%% if the line parallel with the horizontal axis s1 = a; s2 = 1/beta; else s2 = (alpha*(alpha*b - beta*a) + beta)/(alpha^2 + beta^2); s1 = (1 - beta*s2)/alpha; end sol = [s1 s2]; end A MATLAB source code for Example 3.6: clear all; format long; clc; close all; CENTERS=[-6 9; 12 9; -1 -6;-8 5; -7 0; 1]; RADII=[3; 2.5; 2.5;1; 2; 4]; % MM NESTEROV METHOD SOLVING SIB FOR DISKS tic; N=10; p=5; eps_p=1e-6; gamma=.5; eps_g=1e-5; sigma = (eps_p/p)^(1/N); tildesigma = (eps_g/gamma)^(1/N); z=[0 0]; x=[-15 3]; n=size(CENTERS,1); m=size(CENTERS,2); for i=1:n C(i,:)=proj2ball(x,CENTERS(i,:), RADII(i)); end for i=1:N hold on if i==1 plot(x(1),x(2),’k.’); text(x(1)-1.7,x(2)-.7,’$x^0$’, ’Interpreter’,’latex’, ’fontsize’,13) 120 for j=1:n plot(C(j,1),C(j,2),’k.’); plot([x(1) C(j,1)], [x(2) C(j,2)],’k ’); end end if i==2 plot(x(1),x(2),’k.’); text(x(1)+.2,x(2)-1.4,’$x^1$’, ’Interpreter’,’latex’, ’fontsize’,13) for j=1:n scatter(C(j,1),C(j,2),’.r’); plot([x(1) C(j,1)], [x(2) C(j,2)],’k’); end end if p > eps_p for k=0:1000000 L = 2/p; G_MAX = 0; for i=1:n G = sqrt([norm(x-C(i,:))]^2 + p^2); if G>G_MAX G_MAX = G; end end TAU=0; for i=1:n l=norm(x-C(i,:)); TAU = TAU + exp(sqrt(l^2 + p^2)/p - G_MAX/p); end NABLA=0; for i=1:n G=sqrt(norm(x-C(i,:))^2 + p^2) ; LAMBDA=exp(G/p - G_MAX/p)/TAU; NABLA = NABLA + (LAMBDA/G)*(x - C(i,:)); end if norm(NABLA)> gamma y = x-NABLA*(1/L); z = z - (k+1)/(2*L)*NABLA; x = 2/(k+3)*z + (k+1)/(k+3)*y; else %i break end end p=sigma*p; gamma=tildesigma*gamma; for i=1:n, 121 C(i,:)=proj2ball(y,CENTERS(i,:), RADII(i)); end else break end end disp(’ MM NESTEROV RESULTS -’) toc; y MM_NES_VAL = f_val(x,CENTERS,RADII) if m==2 theta = 0:0.005:2*pi; set(gcf,’Color’,[1 1]) for i = 1:n plot(CENTERS(i,1)+ RADII(i)*cos(theta),CENTERS(i,2)+ RADII(i)*sin(theta),’k’); fill(CENTERS(i,1)+RADII(i)*cos(theta),CENTERS(i,2)+RADII(i)*sin(theta),[.9 9]); end plot(x(1),x(2),’r*’); plot(x(1)+ MM_NES_VAL*cos(theta),x(2)+ MM_NES_VAL*sin(theta),’k’,’LineWidth’,1.5); axis off; end % SUBFRADIENT METHOD SOLVING SIB FOR DISKS tic; x=[-15, 3]; V=0; d=0; for i=1:n if V f_max) f_max = temp; end end f = f_max; end % -function proj=proj2ball(x,c,r) if norm(x-c)= -130; NEGATIVE = CITIES(:, 2) < -130; a = CITIES(POSITIVE, :); b = CITIES(NEGATIVE, :); c = [30 -160]; r = 30; %size(a) %size(b) N = 10; M = 10; x = c; x_Weiszfeld = [0 0]; V_Weiszfeld = zeros(N, 1); % positive part % negative part % center and radius of constrained ball for k = 1:N %% DCA - Weiszfeld y = SUBH(x, b) + 2*x; 123 x = WEISZFELD_METHOD(x, a, y); V_Weiszfeld(k) = F_VAL(x, a, b); x_Weiszfeld(k, :) = x; % Weiszfeld Method end fprintf(’\n\n’); disp(’ -OPTIMAL SOLUTION OF WEISZFELD METHOD -’) disp(x); toc; accuaracy = CHECK_ACCUARACY(1000); % check accuracy over 1000 samples fprintf(’Accuracy: %2.2f%s \n\n’, accuaracy, ’%’); clear toc; tic x = [1 1]; x_save_sub = [0 0]; V_sub = zeros(N, 1); for k = 1:N % DCA - Subgradient y = SUBH(x, b) + 2*x; x = SUBGRADIENT_METHOD(x, a, y); % Subgradient method V_sub(k) = F_VAL(x, a, b); x_save_sub(k, :) = x; end toc fprintf(’\n\n’); disp(’ -OPTIMAL SOLUTION OF SUBGRADIENT METHOD -’) disp(x); accuaracy = CHECK_ACCUARACY(1000); fprintf(’Accuracy: %2.2f%s \n\n’, accuaracy, ’%’); error_Weiszfeld = zeros(N-1, 1); for q=1:(N-1) error_Weiszfeld(q) = norm(x_Weiszfeld(q, :) - x_Weiszfeld(q + 1, :)); end error_sub = zeros(N-1, 1); for i=1:(N-1) error_sub(i) = norm(x_save_sub(i, :) - x_save_sub(i + 1, :)); end figure(1) semilogy(1:length(error_Weiszfeld), error_Weiszfeld, ’LineWidth’, 2); hold on; semilogy(1:length(error_sub), error_sub, ’g’, ’LineWidth’, 2); legend(’Weiszfeld’, ’Subgradient’); xlabel(’$k$’, ’Interpreter’,’latex’, ’fontsize’,14); 124 ylabel(’$\|x_{k+1} - x_k\|$’, ’Interpreter’,’latex’, ’fontsize’,14); error_Weiszfeld_val = zeros(N-1, 1); for i=1:N-1 error_Weiszfeld_val(i) = norm(V_Weiszfeld(i) - V_Weiszfeld(i + 1)); end error_sub_val = zeros(N-1, 1); for i=1:N-1 error_sub_val(i) = norm(V_sub(i) - V_sub(i + 1)); end for tt = 1:N fprintf(’[%2.8f, %3.8f]\n’,x_Weiszfeld(tt, 1), x_Weiszfeld(tt, 2)) end disp(’\n’) for tt = 1:N fprintf(’[%2.8f, %3.8f]\n’,x_save_sub(tt, 1), x_save_sub(tt, 2)) end disp(’\n’) for tt = 1:N fprintf(’%.12f\n’,V_Weiszfeld(tt)); end disp(’\n’) for tt = 1:N fprintf(’%.12f\n’,V_sub(tt)); end V_Weiszfeld - V_sub disp(’ - OPTIMAL VALUE OF WEISZFEL METHOD -’) V_Weiszfeld(N) disp(’ - OPTIMAL VALUE OF SUBGRADIENT METHOD -’) V_sub(N) figure(2); DrawResult(); % -function sub = SUBH(x, b) %% compute subdifferential of h(x) sub = 0; for u = 1:size(b, 1) if norm(x - b(u, :))>0 sub = sub + (x - b(u, :))/norm(x - b(u, :)); end end end % -function val = F_VAL(x, a, b) %% compute the value function alpha = g(x) - h(x) 125 val = 0; for u = 1:size(a, 1) val = val + norm(x - a(u, :)); end for v = 1:size(b, 1) val = val - norm(x - b(v, :)); end end % -function z = WEISZFELD_METHOD(x, a, y) %% solving subproblems by Weiszfeld method varphi_y with starting point x z = x; for e=1:M num = 0; den = 0; for g = 1:size(a, 1) aa = a(g, :); %if(z == aa) % z = aa; break; %end num = num + aa/norm(z - aa); den = den + 1/norm(z - aa); end z = (num + y)/(den + 2); if(norm(z - c) > r) z = c + r*(z - c)/norm(z - c); end end end % -function z = SUBGRADIENT_METHOD(x, a, y) %% Subgradient Method z = x; for t = 1:M z = z - 1/t*(2*z + SUBH(z, a) - y); % Subgradient method for sets if(norm(z - c) > r) z = c + r*(z - c)/norm(z - c); end end end % -function accuaracy = CHECK_ACCUARACY(sample) theta = linspace(0,2*pi,200); % create vector theta n = sample; t = 2*pi*rand(n,1); rt = r*sqrt(rand(n,1)); u = c(1) + rt.*cos(t); v = c(2) + rt.*sin(t); 126 X = [u,v]; %size(X) indicator = 0; opti_value = F_VAL(x, a, b); for u = 1:length(X) temp = F_VAL(X(u, :), a, b); if temp > opti_value indicator = indicator + 1; end end accuaracy = indicator/length(X)*100; end % -function DrawResult() DRAW_DISK(c(2), c(1), r, ’bla’); % draw the constraint hold on; for u=1:length(a) plot([a(u, 2) x(1, 2)], [a(u, 1) x(1, 1)], ’g’) hold on; end for v=1:length(b) plot([b(v, 2) x(1, 2)], [b(v, 1) x(1, 1)], ’blu’) hold on; end for p=1:length(a) % plot points a plot(a(p, 2), a(p, 1), ’blu.’, ’markersize’, 12) hold on; end for s=1:length(b) % plot points b plot(b(s, 2), b(s, 1), ’m.’, ’markersize’, 12) hold on; end plot(x(1, 2), x(1, 1), ’r.’, ’markersize’, 15) xlabel(’$\mathrm{Longitude}$’, ’Interpreter’,’latex’, ’fontsize’,14); ylabel(’$\mathrm{Latitude}$’, ’Interpreter’,’latex’, ’fontsize’,14); end end % -function DRAW_DISK(cx,cy,r,t) format long % makes numerical output in double precision theta = linspace(0,2*pi,200); % create vector theta x = cx + r * cos(theta); % generate x-coordinate y = cy + r * sin(theta); % generate y-coordinate plot(x,y,’Color’,t,’Linewidth’,2); % plot circle axis(’equal’); % set equal scale on axes per pixel end 127 Index algorithm DCA, 10 generalized proximal point, 93 generalized Weiszfeld, 80 majorization-minimization, Nesterov’s accelerated gradient, Weiszfeld, 26 convex, Fr´echet, 87 limiting, 87 uniqueness of solutions, 18, 24, 40, 41, 76 function strictly convex, convex, Euclidean distance, extended-real-valued, generalized distance, 12 indicator, log-exponential smoothing, 48 proper, strongly convex, support, 12 Kurdyka - Lojasiewicz property, 89 majorization-minimization principle, normally round, 22 normally smooth, 17 optimality condition, 9, 41, 92 problem classical Fermat-Torricelli, 11 convex feasibility, 53 generalized Fermat-Torricelli, 12 nonconvex feasibility, 104 nonconvex Fermat-Torricelli, 65 smallest enclosing ball, 36 smallest enclosing circle, 36 smallest intersecting ball, 37 support vector machine, 55 projection Euclidean, generalized, 21 strongly convex, subdifferential Clarke, 88 128