[4] | 1 | %% Constrained optimization with FMINSEARCH |
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| 2 | |
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| 3 | function testoptimize |
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| 4 | |
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| 5 | |
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| 6 | %% Introduction |
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| 7 | |
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| 8 | %% Usage |
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| 9 | |
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| 10 | %% |
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| 11 | % first, define a test function: |
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| 12 | clc, rosen = @(x) (1-x(1))^2 + 105*(x(2)-x(1)^2)^2; |
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| 13 | |
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| 14 | %% |
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| 15 | % this is the classical Rosenbr\"uck function, which has a global minimum |
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| 16 | % at _f(x)_ = _f_([1, 1]) = 0. The function is relatively hard to optimize, |
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| 17 | % because that minimum is located in a long narrow ``valley'': |
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| 18 | k = 0; range = -5:0.1:5; |
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| 19 | z = zeros(101); |
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| 20 | for i = range |
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| 21 | m = 0; k = k + 1; |
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| 22 | for j = range |
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| 23 | m = m + 1; |
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| 24 | z(k, m) = rosen([i, j]); |
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| 25 | end |
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| 26 | end |
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| 27 | [y, x] = meshgrid(range, range); |
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| 28 | surf(x, y, z, 'linestyle', 'none'), view(-150, 30), axis tight |
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| 29 | |
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| 30 | %% |
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| 31 | % Optimizing the fully unconstrained problem with OPTIMIZE indeed finds |
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| 32 | % the global minimum: |
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| 33 | % warning work only with optimization toolbox !!! |
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| 34 | |
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| 35 | solution = optimize(rosen, [3 3]) |
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| 36 | |
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| 37 | %% |
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| 38 | % Imposing a lower bound on the variables gives |
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| 39 | [solution, fval] = optimize(rosen, [3 3], [2 2], []) |
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| 40 | |
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| 41 | %% |
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| 42 | % in the figure, this looks like |
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| 43 | zz = z; zz(x > 2 & y > 2) = inf; |
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| 44 | ZZ = z; ZZ(x < 2) = inf; ZZ(y < 2) = inf; |
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| 45 | figure, hold on |
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| 46 | surf(x, y, zz, 'linestyle', 'none', 'FaceAlpha', 0.2) |
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| 47 | surf(x, y, ZZ, 'linestyle', 'none') |
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| 48 | view(-150, 30), grid on, axis tight |
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| 49 | plot3(solution(1), solution(2), fval+1e3, 'g.', 'MarkerSize', 20) |
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| 50 | |
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| 51 | %% |
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| 52 | % Similarly, imposing an upper bound yields |
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| 53 | solution = optimize(rosen, [3 3], [], [0.5 0.5]) |
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| 54 | |
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| 55 | zz = z; zz(x < 0.5 & y < 0.5) = inf; |
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| 56 | ZZ = z; ZZ(x > 0.5) = inf; ZZ(y > 0.5) = inf; |
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| 57 | figure, hold on |
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| 58 | surf(x, y, zz, 'linestyle', 'none', 'FaceAlpha', 0.2) |
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| 59 | surf(x, y, ZZ, 'linestyle', 'none') |
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| 60 | view(150, 30), grid on, axis tight |
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| 61 | plot3(solution(1), solution(2), fval+1e3, 'g.', 'MarkerSize', 20) |
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| 62 | |
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| 63 | %% |
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| 64 | % Optimize with _x_(2) fixed at 3. In this case, OPTIMIZE simply |
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| 65 | % removes the variable before FMINSEARCH sees it, essentially |
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| 66 | % reducing the dimensionality of the problem. This is particularly |
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| 67 | % useful when the number of dimensions _N_ becomes large. |
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| 68 | optimize(rosen, [3 3], [-inf 3], [inf 3]) |
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| 69 | |
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| 70 | %% |
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| 71 | % Also general nonlinear constraints can be used. A simple example: |
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| 72 | % |
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| 73 | % nonlinear inequality: |
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| 74 | % |
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| 75 | % $$\sqrt{x_1^2 + x_2^2} \leq 1$$ |
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| 76 | % |
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| 77 | % nonlinear equality : |
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| 78 | % |
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| 79 | % $$x_1^2 + x_2^3 = 0.5$$ |
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| 80 | |
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| 81 | options = optimset('TolFun', 1e-8, 'TolX', 1e-8); |
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| 82 | [sol, fval, exitflag, output] = optimize(rosen, [3 -3], [], [], [], ... |
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| 83 | [], [], [], @nonlcon, [], options); |
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| 84 | |
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| 85 | %% |
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| 86 | % Note that |nonlcon| is a subfunction, listed below. In a figure, this |
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| 87 | % looks like |
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| 88 | zz = z; zz(sqrt(x.^2 + y.^2) <= 1) = inf; |
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| 89 | ZZ = z; ZZ(sqrt(x.^2 + y.^2) >= 1.2) = inf; |
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| 90 | zZ = z; zZ(x.^2 + y.^3 >= 0.5 + 0.1) = inf; |
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| 91 | zZ(x.^2 + y.^3 <= 0.5 - 0.1) = inf; |
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| 92 | figure, hold on |
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| 93 | surf(x, y, zz, 'linestyle', 'none', 'FaceAlpha', 0.2) |
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| 94 | surf(x, y, ZZ, 'linestyle', 'none') |
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| 95 | xX = x(isfinite(zZ)); xX = xX(:); |
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| 96 | yY = y(isfinite(zZ)); xX = xX(:); |
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| 97 | zZ = zZ(isfinite(zZ)); zZ = zZ(:); |
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| 98 | [xX, inds] = sort(xX); yY = yY(inds); zZ = zZ(inds); |
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| 99 | xyz = [xX, yY, zZ]; |
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| 100 | for i = 1:length(xX)-1 % line-command is *somewhat* inconvenient... |
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| 101 | l = line( [xyz(i, 1); xyz(i+1, 1)],[xyz(i, 2); xyz(i+1, 2)], [xyz(i, 3); xyz(i+1, 3)]); |
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| 102 | set(l, 'color', 'r', 'linewidth', 2) |
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| 103 | end |
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| 104 | view(150, 50), grid on, axis tight |
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| 105 | plot3(sol(1), sol(2), fval+1e3, 'g.', 'MarkerSize', 20) |
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| 106 | |
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| 107 | %% |
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| 108 | % Note that the output structure contains a field ``constrviolation'': |
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| 109 | output |
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| 110 | |
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| 111 | %% |
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| 112 | % The contents of which shows that all constraints have been satisfied: |
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| 113 | output.constrviolation |
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| 114 | |
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| 115 | |
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| 116 | end |
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| 117 | |
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| 118 | %% |
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| 119 | function [c, ceq] = nonlcon(x) |
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| 120 | c = norm(x) - 1; |
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| 121 | ceq = x(1)^2 + x(2)^3 - 0.5; |
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| 122 | end |
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| 123 | |
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