| Developer(s) | Tomlab Optimization Inc. |
|---|---|
| Stable release | 7.4 / March 24, 2010 |
| Operating system | TOMLAB - OS Support |
| Type | Technical computing |
| License | Proprietary |
| Website | TomSym product page |
The TomSym[1] MATLAB symbolic modeling engine is a platform for modeling applied optimization and optimal control problems.
Contents |
TomSym is complete modeling environment in Matlab with support for most built-in mathematical operators in Matlab. It is a combined modeling, compilation and interface to the TOMLAB solvers. The matrix derivative of a matrix function is a fourth rank tensor - that is, a matrix each of whose entries is a matrix. Rather than using four-dimensional matrices to represent this, TomSym continues to work in two dimensions. This makes it possible to take advantage of the very efficient handling of sparse matrices in Matlab, which is not available for higher-dimensional matrices.
TomSym has a variety of functions, among them:
The TomSym symbolic source transformation makes it possible to define any the set of decision variables (both continuous and integer) and any type of constraint as well as scalars and constant parameters.
An example linear programming problem would look like this:
c = [-7; -5]; A = [ 1 2 4 1 ]; b_U = [ 6; 12 ]; x_L = [ 0; 0 ]; toms 2x1 x solution = ezsolve(c'*x, {A*x<=b_U, x_L<=x});
A MINLP problem is defined just like a linear programming problem. This example also shows how to convert the model into a general TOMLAB problem.
Name='minlp1Demo - Kocis/Grossman.'; toms 2x1 x toms 3x1 integer y objective = [2 3 1.5 2 -0.5]*[x;y]; constraints = { ... x(1) >= 0, ... x(2) >= 1e-8, ... x <= 1e8, ... 0 <= y <=1, ... [1 0 1 0 0]*[x;y] <= 1.6, ... 1.333*x(2) + y(2) <= 3, ... [-1 -1 1]*y <= 0, ... x(1)^2+y(1) == 1.25, ... sqrt(x(2)^3)+1.5*y(2) == 3, ... }; guess = struct('x',ones(size(x)),'y',ones(size(y))); options = struct; options.name = Name; Prob = sym2prob('minlp',objective,constraints,guess,options); Prob.DUNDEE.optPar(20) = 1; Result = tomRun('minlpBB',Prob,2);
tomSym makes it possible to build models with two or more variable indices in MATLAB[2]. The following example creates a variable 'flow' with four indices. The variable is then used to create a constraint over two of the indices and to sum the multiplication with a two-dimensional matrix.
% Create the indices used in model i = tomArrayIdx('i',1:6); j = tomArrayIdx('j',1:6); k = tomArrayIdx('k',1:6); l = tomArrayIdx('l',1:6); % Create an integer variable of full length flow = tom('flow',6^4,1,'int'); % Convert the variable to a matrix with four indices. flow = tomArray(flow,[6,6,6,6]); % Create a constraint valid for all i and j cons = {sum(sum(flow(i,j,k,l),k),l) == 1}; % Create a scalar based on multi-index multiplications distance = tomArray([ 0 945 605 4667 4749 4394;... 945 0 866 3726 3806 3448;... 605 866 0 4471 4541 4152;... 4667 3726 4471 0 109 415;... 4749 3806 4541 109 0 431;... 4394 3448 4152 415 431 0]); sumtotal = sum(vec((distance(i,k)+distance(l,j)+... distance(k,l)*.8).*flow(i,j,k,l)));
For functions that cannot be interpreted by tomSym it is possible to use either automatic differentiation or numerical differentiation. In the following example a simple problem is solved using the two methods.
toms x1 x2 alpha = 100; % USE MAD (AUTOMATIC DIFFERENTIATION) FOR ONE FUNCTION % % Create a wrapper function. In this case we use sin, but it could be any % MAD supported function. y = wrap(struct('fun','sin','n',1,'sz1',1,'sz2',1,'JFuns','MAD'),x1/x2); f = alpha*(x2-x1^2)^2 + (1-x1)^2 + y; % Setup and solve the problem c = -x1^2 - x2; con = {-1000 <= c <= 0 -10 <= x1 <= 2 -10 <= x2 <= 2}; x0 = {x1 == -1.2 x2 == 1}; solution1 = ezsolve(f,con,x0); % USE NUMERICAL DIFFERENTIATION FOR ONE FUNCTIONS % Create a new wrapper function. In this case we use sin, but it could be % any function since we use numerical derivatives. y = wrap(struct('fun','sin','n',1,'sz1',1,'sz2',1,'JFuns','FDJac'),x1/x2); f = alpha*(x2-x1^2)^2 + (1-x1)^2 + y; solution2 = ezsolve(f,con,x0);