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If you have a Symbolic Math Toolbox™ license, you can easily calculate analytic gradients and Hessians for objective and constraint functions. There are two relevant Symbolic Math Toolbox functions:
jacobian generates the gradient of a scalar function, and generates a matrix of the partial derivatives of a vector function. So, for example, you can obtain the Hessian matrix, the second derivatives of the objective function, by applying jacobian to the gradient. Part of this example shows how to use jacobian to generate symbolic gradients and Hessians of objective and constraint functions.
matlabFunction generates either an anonymous function or a file that calculates the values of a symbolic expression. This example shows how to use matlabFunction to generate files that evaluate the objective and constraint function and their derivatives at arbitrary points.
Consider the electrostatics problem of placing 10 electrons in a conducting body. The electrons will arrange themselves so as to minimize their total potential energy, subject to the constraint of lying inside the body. It is well known that all the electrons will be on the boundary of the body at a minimum. The electrons are indistinguishable, so there is no unique minimum for this problem (permuting the electrons in one solution gives another valid solution). This example was inspired by Dolan, Moré, and Munson [58].
This example is a conducting body defined by the following inequalities:
This body looks like a pyramid on a sphere.
There is a slight gap between the upper and lower surfaces of the figure. This is an artifact of the general plotting routine used to create the figure. This routine erases any rectangular patch on one surface that touches the other surface.
The syntax and structures of the two sets of toolbox functions differ. In particular, symbolic variables are real or complex scalars, but Optimization Toolbox™ functions pass vector arguments. So there are several steps to take to generate symbolically the objective function, constraints, and all their requisite derivatives, in a form suitable for the interior-point algorithm of fmincon:
To see the efficiency in using gradients and Hessians, see Compare to Optimization Without Gradients and Hessians.
Generate a symbolic vector x as a 30-by-1 vector composed of real symbolic variables xij, i between 1 and 10, and j between 1 and 3. These variables represent the three coordinates of electron i: xi1 corresponds to the x coordinate, xi2 corresponds to the y coordinate, and xi3 corresponds to the z coordinate.
x = cell(3, 10); for i = 1:10 for j = 1:3 x{j,i} = sprintf('x%d%d',i,j); end end x = x(:); % now x is a 30-by-1 vector x = sym(x, 'real');
The vector x is:
x x = x11 x12 x13 x21 x22 x23 x31 x32 x33 x41 x42 x43 x51 x52 x53 x61 x62 x63 x71 x72 x73 x81 x82 x83 x91 x92 x93 x101 x102 x103
Write the linear constraints in Equation 6-60,
z ≤ –|x| – |y|,
as a set of four linear inequalities for each electron:
xi3 – xi1 – xi2 ≤
0
xi3 – xi1 + xi2 ≤
0
xi3 + xi1 – xi2 ≤
0
xi3 + xi1 + xi2 ≤
0
Therefore there are a total of 40 linear inequalities for this problem.
Write the inequalities in a structured way:
B = [1,1,1;-1,1,1;1,-1,1;-1,-1,1]; A = zeros(40,30); for i=1:10 A(4*i-3:4*i,3*i-2:3*i) = B; end b = zeros(40,1);
You can see that A*x ≤ b represents the inequalities:
A*x ans = x11 + x12 + x13 x12 - x11 + x13 x11 - x12 + x13 x13 - x12 - x11 x21 + x22 + x23 x22 - x21 + x23 x21 - x22 + x23 x23 - x22 - x21 x31 + x32 + x33 x32 - x31 + x33 x31 - x32 + x33 x33 - x32 - x31 x41 + x42 + x43 x42 - x41 + x43 x41 - x42 + x43 x43 - x42 - x41 x51 + x52 + x53 x52 - x51 + x53 x51 - x52 + x53 x53 - x52 - x51 x61 + x62 + x63 x62 - x61 + x63 x61 - x62 + x63 x63 - x62 - x61 x71 + x72 + x73 x72 - x71 + x73 x71 - x72 + x73 x73 - x72 - x71 x81 + x82 + x83 x82 - x81 + x83 x81 - x82 + x83 x83 - x82 - x81 x91 + x92 + x93 x92 - x91 + x93 x91 - x92 + x93 x93 - x92 - x91 x101 + x102 + x103 x102 - x101 + x103 x101 - x102 + x103 x103 - x102 - x101
The nonlinear constraints in Equation 6-61 ,
$${x}^{2}+{y}^{2}+{\left(z+1\right)}^{2}\le 1,$$
are also structured. Generate the constraints, their gradients, and Hessians as follows:
c = sym(zeros(1,10)); i = 1:10; c = (x(3*i-2).^2 + x(3*i-1).^2 + (x(3*i)+1).^2 - 1).'; gradc = jacobian(c,x).'; % .' performs transpose hessc = cell(1, 10); for i = 1:10 hessc{i} = jacobian(gradc(:,i),x); end
The constraint vector c is a row vector, and the gradient of c(i) is represented in the ith column of the matrix gradc. This is the correct form, as described in Nonlinear Constraints.
The Hessian matrices, hessc{1}...hessc{10}, are square and symmetric. It is better to store them in a cell array, as is done here, than in separate variables such as hessc1, ..., hesssc10.
Use the .' syntax to transpose. The ' syntax means conjugate transpose, which has different symbolic derivatives.
The objective function, potential energy, is the sum of the inverses of the distances between each electron pair:
$$\text{energy}={\displaystyle \sum _{i<j}\frac{1}{\left|{x}_{i}-{x}_{j}\right|}}.$$
The distance is the square root of the sum of the squares of the differences in the components of the vectors.
Calculate the energy, its gradient, and its Hessian as follows:
energy = sym(0); for i = 1:3:25 for j = i+3:3:28 dist = x(i:i+2) - x(j:j+2); energy = energy + 1/sqrt(dist.'*dist); end end gradenergy = jacobian(energy,x).'; hessenergy = jacobian(gradenergy,x);
The objective function should have two outputs, energy and gradenergy. Put both functions in one vector when calling matlabFunction to reduce the number of subexpressions that matlabFunction generates, and to return the gradient only when the calling function (fmincon in this case) requests both outputs. This example shows placing the resulting files in your current folder. Of course, you can place them anywhere you like, as long as the folder is on the MATLAB path.
currdir = [pwd filesep]; % You may need to use currdir = pwd filename = [currdir,'demoenergy.m']; matlabFunction(energy,gradenergy,'file',filename,'vars',{x});
This syntax causes matlabFunction to return energy as the first output, and gradenergy as the second. It also takes a single input vector {x} instead of a list of inputs x11, ..., x103.
The resulting file demoenergy.m contains, in part, the following lines or similar ones:
function [energy,gradenergy] = demoenergy(in1) %DEMOENERGY % [ENERGY,GRADENERGY] = DEMOENERGY(IN1) ... x101 = in1(28,:); ... energy = 1./t140.^(1./2) + ...; if nargout > 1 ... gradenergy = [(t174.*(t185 - 2.*x11))./2 - ...]; end
This function has the correct form for an objective function with a gradient; see Writing Scalar Objective Functions.
Generate the nonlinear constraint function, and put it in the correct format.
filename = [currdir,'democonstr.m']; matlabFunction(c,[],gradc,[],'file',filename,'vars',{x},... 'outputs',{'c','ceq','gradc','gradceq'});
The resulting file democonstr.m contains, in part, the following lines or similar ones:
function [c,ceq,gradc,gradceq] = democonstr(in1) %DEMOCONSTR % [C,CEQ,GRADC,GRADCEQ] = DEMOCONSTR(IN1) ... x101 = in1(28,:); ... c = [t417.^2 + ...]; if nargout > 1 ceq = []; end if nargout > 2 gradc = [2.*x11,...]; end if nargout > 3 gradceq = []; end
This function has the correct form for a constraint function with a gradient; see Nonlinear Constraints.
To generate the Hessian of the Lagrangian for the problem, first generate files for the energy Hessian and for the constraint Hessians.
The Hessian of the objective function, hessenergy, is a very large symbolic expression, containing over 150,000 symbols, as evaluating size(char(hessenergy)) shows. So it takes a substantial amount of time to run matlabFunction(hessenergy).
To generate a file hessenergy.m, run the following two lines:
filename = [currdir,'hessenergy.m']; matlabFunction(hessenergy,'file',filename,'vars',{x});
In contrast, the Hessians of the constraint functions are small, and fast to compute:
for i = 1:10 ii = num2str(i); thename = ['hessc',ii]; filename = [currdir,thename,'.m']; matlabFunction(hessc{i},'file',filename,'vars',{x}); end
After generating all the files for the objective and constraints, put them together with the appropriate Lagrange multipliers in a file hessfinal.m as follows:
function H = hessfinal(X,lambda) % % Call the function hessenergy to start H = hessenergy(X); % Add the Lagrange multipliers * the constraint Hessians H = H + hessc1(X) * lambda.ineqnonlin(1); H = H + hessc2(X) * lambda.ineqnonlin(2); H = H + hessc3(X) * lambda.ineqnonlin(3); H = H + hessc4(X) * lambda.ineqnonlin(4); H = H + hessc5(X) * lambda.ineqnonlin(5); H = H + hessc6(X) * lambda.ineqnonlin(6); H = H + hessc7(X) * lambda.ineqnonlin(7); H = H + hessc8(X) * lambda.ineqnonlin(8); H = H + hessc9(X) * lambda.ineqnonlin(9); H = H + hessc10(X) * lambda.ineqnonlin(10);
Start the optimization with the electrons distributed randomly on a sphere of radius 1/2 centered at [0,0,–1]:
rng default % for reproducibility Xinitial = randn(3,10); % columns are normal 3-D vectors for j=1:10 Xinitial(:,j) = Xinitial(:,j)/norm(Xinitial(:,j))/2; % this normalizes to a 1/2-sphere end Xinitial(3,:) = Xinitial(3,:) - 1; % center at [0,0,-1] Xinitial = Xinitial(:); % Convert to a column vector
Set the options to use the interior-point algorithm, and to use gradients and the Hessian:
options = optimoptions(@fmincon,'Algorithm','interior-point','GradObj','on',... 'GradConstr','on','Hessian','user-supplied',... 'HessFcn',@hessfinal,'Display','final');
Call fmincon:
[xfinal fval exitflag output] = fmincon(@demoenergy,Xinitial,... A,b,[],[],[],[],@democonstr,options)
The output is as follows:
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the default value of the function tolerance, and constraints are satisfied to within the default value of the constraint tolerance. xfinal = -0.0317 0.0317 -1.9990 0.6356 -0.6356 -1.4381 0.0000 -0.0000 -0.0000 0.0000 -1.0000 -1.0000 1.0000 -0.0000 -1.0000 -1.0000 -0.0000 -1.0000 0.6689 0.6644 -1.3333 -0.6667 0.6667 -1.3333 0.0000 1.0000 -1.0000 -0.6644 -0.6689 -1.3333 fval = 34.1365 exitflag = 1 output = iterations: 19 funcCount: 28 constrviolation: 0 stepsize: 4.0372e-05 algorithm: 'interior-point' firstorderopt: 4.0015e-07 cgiterations: 55 message: 'Local minimum found that satisfies the constraints. Optimizati...'
Even though the initial positions of the electrons were random, the final positions are nearly symmetric:
The use of gradients and Hessians makes the optimization run faster and more accurately. To compare with the same optimization using no gradient or Hessian information, set the options not to use gradients and Hessians:
options = optimoptions(@fmincon,'Algorithm','interior-point',... 'Display','final'); [xfinal2 fval2 exitflag2 output2] = fmincon(@demoenergy,Xinitial,... A,b,[],[],[],[],@democonstr,options)
The output shows that fmincon found an equivalent minimum, but took more iterations and many more function evaluations to do so:
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the default value of the function tolerance, and constraints are satisfied to within the default value of the constraint tolerance. xfinal2 = 0.0000 1.0000 -1.0000 0.6689 -0.6644 -1.3334 -0.6644 0.6689 -1.3334 0.0000 -1.0000 -1.0000 0.6357 0.6357 -1.4380 -0.0317 -0.0317 -1.9990 1.0000 0.0000 -1.0000 -1.0000 0.0000 -1.0000 0.0000 0.0000 -0.0000 -0.6667 -0.6667 -1.3334 fval2 = 34.1365 exitflag2 = 1 output2 = iterations: 77 funcCount: 2431 constrviolation: 0 stepsize: 6.0588e-07 algorithm: 'interior-point' firstorderopt: 2.9894e-06 cgiterations: 0 message: 'Local minimum found that satisfies the constraints. Optimizati...'
In this run the number of function evaluations (in output2.funcCount) is 2431, compared to 28 (in output.funcCount) when using gradients and Hessian.
The symbolic variables in this example have the assumption, in the symbolic engine workspace, that they are real. To clear this assumption from the symbolic engine workspace, it is not sufficient to delete the variables. You must clear the variables using the syntax
syms x11 x12 x13 clear
or reset the symbolic engine using the command
reset(symengine)
After resetting the symbolic engine you should clear all symbolic variables from the MATLAB workspace with the clear command, or clear variable_list.