100 variable problems are hugely more complex than 3 variable problems on a very limited search space. You can often solve the small problem using an exhaustive search, but the curse of dimensionality kills you if you go higher.
Anyway, you still have not told us about some significant details of your real problem. If is it linear, and any constraints are also linear, then use intlinprog. Use a linear integer solver to solve linear integer problems.
help intlinprog
INTLINPROG Mixed integer linear programming.
X = INTLINPROG(f,intcon,A,b) attempts to solve problems of the form
min f'*x subject to: A*x <= b
x Aeq*x = beq
lb <= x <= ub
x(i) integer, where i is in the index
vector intcon (integer constraints)
X = INTLINPROG(f,intcon,A,b) solves the problem with integer variables
in the intcon vector and linear inequality constraints A*x <= b. intcon
is a vector of positive integers indicating components of the solution
X that must be integers. For example, if you want to constrain X(2) and
X(10) to be integers, set intcon to [2,10].
X = INTLINPROG(f,intcon,A,b,Aeq,beq) solves the problem above while
additionally satisfying the equality constraints Aeq*x = beq. (Set A=[]
and b=[] if no inequalities exist.)
X = INTLINPROG(f,intcon,A,b,Aeq,beq,LB,UB) defines a set of lower and
upper bounds on the design variables, X, so that the solution is in the
range LB <= X <= UB. Use empty matrices for LB and UB if no bounds
exist. Set LB(i) = -Inf if X(i) is unbounded below; set UB(i) = Inf if
X(i) is unbounded above.
X = INTLINPROG(f,intcon,A,b,Aeq,beq,LB,UB,X0) sets the initial point
to X0.
X = INTLINPROG(f,intcon,A,b,Aeq,beq,LB,UB,X0,OPTIONS) minimizes with
the default optimization parameters replaced by values in OPTIONS, an
argument created with the OPTIMOPTIONS function. See OPTIMOPTIONS for
details.
X = INTLINPROG(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a
structure with the vector 'f' in PROBLEM.f, the integer constraints in
PROBLEM.intcon, the linear inequality constraints in PROBLEM.Aineq and
PROBLEM.bineq, the linear equality constraints in PROBLEM.Aeq and
PROBLEM.beq, the lower bounds in PROBLEM.lb, the upper bounds in
PROBLEM.ub, the initial point in PROBLEM.x0, the options structure in
PROBLEM.options, and solver name 'intlinprog' in PROBLEM.solver.
[X,FVAL] = INTLINPROG(f,intcon,A,b,...) returns the value of the
objective function at X: FVAL = f'*X.
[X,FVAL,EXITFLAG] = INTLINPROG(f,intcon,A,b,...) returns an EXITFLAG
that describes the exit condition. Possible values of EXITFLAG and the
corresponding exit conditions are
3 Optimal solution found with poor constraint feasibility.
2 Solver stopped prematurely. Integer feasible point found.
1 Optimal solution found.
0 Solver stopped prematurely. No integer feasible point found.
-1 Solver stopped by an output function or plot function.
-2 No feasible point found.
-3 Root LP problem is unbounded.
-9 Solver lost feasibility probably due to ill-conditioned matrix.
[X,FVAL,EXITFLAG,OUTPUT] = INTLINPROG(f,A,b,...) returns a structure
OUTPUT containing information about the optimization process. OUTPUT
includes the number of integer feasible points found and the final gap
between internally calculated bounds on the solution. See the
documentation for a complete description.
See also LINPROG.
Documentation for intlinprog
doc intlinprog
If your problem is not of that kind, then you CANNOT use fmincon as you say you wanted. fmincon absolutely requires a continuous domain and a continuous objective. A smooth objective too.
For the problem you describe, you probably need to use GA from the global optimization TB.
help ga
GA Constrained optimization using genetic algorithm.
GA attempts to solve problems of the following forms:
min F(X) subject to: A*X <= B, Aeq*X = Beq (linear constraints)
X C(X) <= 0, Ceq(X) = 0 (nonlinear constraints)
LB <= X <= UB
X(i) integer, where i is in the index
vector INTCON (integer constraints)
Note: If INTCON is not empty, then no equality constraints are allowed.
That is:-
* Aeq and Beq must be empty
* Ceq returned from NONLCON must be empty
X = GA(FITNESSFCN,NVARS) finds a local unconstrained minimum X to the
FITNESSFCN using GA. NVARS is the dimension (number of design
variables) of the FITNESSFCN. FITNESSFCN accepts a vector X of size
1-by-NVARS, and returns a scalar evaluated at X.
X = GA(FITNESSFCN,NVARS,A,b) finds a local minimum X to the function
FITNESSFCN, subject to the linear inequalities A*X <= B. Linear
constraints are not satisfied when the PopulationType option is set to
'bitString' or 'custom'. See the documentation for details.
X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq) finds a local minimum X to the
function FITNESSFCN, subject to the linear equalities Aeq*X = beq as
well as A*X <= B. (Set A=[] and B=[] if no inequalities exist.) Linear
constraints are not satisfied when the PopulationType option is set to
'bitString' or 'custom'. See the documentation for details.
X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub) defines a set of lower and
upper bounds on the design variables, X, so that a solution is found in
the range lb <= X <= ub. Use empty matrices for lb and ub if no bounds
exist. Set lb(i) = -Inf if X(i) is unbounded below; set ub(i) = Inf if
X(i) is unbounded above. Linear constraints are not satisfied when the
PopulationType option is set to 'bitString' or 'custom'. See the
documentation for details.
X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub,NONLCON) subjects the
minimization to the constraints defined in NONLCON. The function
NONLCON accepts X and returns the vectors C and Ceq, representing the
nonlinear inequalities and equalities respectively. GA minimizes
FITNESSFCN such that C(X)<=0 and Ceq(X)=0. (Set lb=[] and/or ub=[] if
no bounds exist.) Nonlinear constraints are not satisfied when the
PopulationType option is set to 'bitString' or 'custom'. See the
documentation for details.
X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub,NONLCON,options) minimizes
with the default optimization parameters replaced by values in OPTIONS.
OPTIONS can be created with the OPTIMOPTIONS function. See OPTIMOPTIONS
for details. For a list of options accepted by GA refer to the
documentation.
X = GA(FITNESSFCN,NVARS,A,b,[],[],lb,ub,NONLCON,INTCON) requires that
the variables listed in INTCON take integer values. Note that GA does
not solve problems with integer and equality constraints. Pass empty
matrices for the Aeq and beq inputs if INTCON is not empty.
X = GA(FITNESSFCN,NVARS,A,b,[],[],lb,ub,NONLCON,INTCON,options)
minimizes with integer constraints and the default optimization
parameters replaced by values in OPTIONS. OPTIONS can be created with
the OPTIMOPTIONS function. See OPTIMOPTIONS for details.
X = GA(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a structure
that has the following fields:
fitnessfcn: <Fitness function>
nvars: <Number of design variables>
Aineq: <A matrix for inequality constraints>
bineq: <b vector for inequality constraints>
Aeq: <Aeq matrix for equality constraints>
beq: <beq vector for equality constraints>
lb: <Lower bound on X>
ub: <Upper bound on X>
nonlcon: <Nonlinear constraint function>
intcon: <Index vector for integer variables>
options: <Options created with optimoptions('ga',...)>
rngstate: <State of the random number generator>
[X,FVAL] = GA(FITNESSFCN, ...) returns FVAL, the value of the fitness
function FITNESSFCN at the solution X.
[X,FVAL,EXITFLAG] = GA(FITNESSFCN, ...) returns EXITFLAG which
describes the exit condition of GA. Possible values of EXITFLAG and the
corresponding exit conditions are
1 Average change in value of the fitness function over
options.MaxStallGenerations generations less than
options.FunctionTolerance and constraint violation less than
options.ConstraintTolerance.
3 The value of the fitness function did not change in
options.MaxStallGenerations generations and constraint violation
less than options.ConstraintTolerance.
4 Magnitude of step smaller than machine precision and constraint
violation less than options.ConstraintTolerance. This exit
condition applies only to nonlinear constraints.
5 Fitness limit reached and constraint violation less than
options.ConstraintTolerance.
0 Maximum number of generations exceeded.
-1 Optimization terminated by the output or plot function.
-2 No feasible point found.
-4 Stall time limit exceeded.
-5 Time limit exceeded.
[X,FVAL,EXITFLAG,OUTPUT] = GA(FITNESSFCN, ...) returns a
structure OUTPUT with the following information:
rngstate: <State of the random number generator before GA started>
generations: <Total generations, excluding HybridFcn iterations>
funccount: <Total function evaluations>
maxconstraint: <Maximum constraint violation>, if any
message: <GA termination message>
[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = GA(FITNESSFCN, ...) returns the
final POPULATION at termination.
[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORES] = GA(FITNESSFCN, ...) returns
the SCORES of the final POPULATION.
Example:
Unconstrained minimization of Rastrigins function:
function scores = myRastriginsFcn(pop)
scores = 10.0 * size(pop,2) + sum(pop.^2 - 10.0*cos(2*pi .* pop),2);
numberOfVariables = 2
x = ga(@myRastriginsFcn,numberOfVariables)
Display plotting functions while GA minimizes
options = optimoptions('ga','PlotFcn',...
{@gaplotbestf,@gaplotbestindiv,@gaplotexpectation,@gaplotstopping});
[x,fval,exitflag,output] = ga(fitfcn,2,[],[],[],[],[],[],[],options)
An example with inequality constraints and lower bounds
A = [1 1; -1 2; 2 1]; b = [2; 2; 3]; lb = zeros(2,1);
fitfcn = @(x)0.5*x(1)^2 + x(2)^2 -x(1)*x(2) -2*x(1) - 6.0*x(2);
% Use mutation function which can handle constraints
options = optimoptions('ga','MutationFcn',@mutationadaptfeasible);
[x,fval,exitflag] = ga(fitfcn,2,A,b,[],[],lb,[],[],options);
If FITNESSFCN or NONLCON are parameterized, you can use anonymous
functions to capture the problem-dependent parameters. Suppose you want
to minimize the fitness given in the function myfit, subject to the
nonlinear constraint myconstr, where these two functions are
parameterized by their second argument a1 and a2, respectively. Here
myfit and myconstr are MATLAB file functions such as
function f = myfit(x,a1)
f = exp(x(1))*(4*x(1)^2 + 2*x(2)^2 + 4*x(1)*x(2) + 2*x(2) + a1);
and
function [c,ceq] = myconstr(x,a2)
c = [1.5 + x(1)*x(2) - x(1) - x(2);
-x(1)*x(2) - a2];
% No nonlinear equality constraints:
ceq = [];
To optimize for specific values of a1 and a2, first assign the values
to these two parameters. Then create two one-argument anonymous
functions that capture the values of a1 and a2, and call myfit and
myconstr with two arguments. Finally, pass these anonymous functions to
GA:
a1 = 1; a2 = 10; % define parameters first
% Mutation function for constrained minimization
options = optimoptions('ga','MutationFcn',@mutationadaptfeasible);
x = ga(@(x)myfit(x,a1),2,[],[],[],[],[],[],@(x)myconstr(x,a2),options)
Example: Solving a mixed-integer optimization problem
An example of optimizing a function where a subset of the variables are
required to be integers:
% Define the objective and call GA. Here variables x(2) and x(3) will
% be integer.
fun = @(x) (x(1) - 0.2)^2 + (x(2) - 1.7)^2 + (x(3) -5.1)^2;
x = ga(fun,3,[],[],[],[],[],[],[],[2 3])
See also OPTIMOPTIONS, FITNESSFUNCTION, GAOUTPUTFCNTEMPLATE, PATTERNSEARCH, @.
Documentation for ga
doc ga
I'm not sure if any of the other solvers in that toolbox are able to handle large problems with integer variables, but GA would be your first choice in any case.