Hi @Melika Eft
The constrained optimization problem probably can be formulated like this:
fun = @(x) x(1)^2 + x(2)^2;
x0 = [1, 2]; % initial guess
A = [-1, 0]; % A*x ≤ b
b = 2;
Aeq = [1, 1]; % Aeq*x = beq
beq = 3;
x = fmincon(fun, x0, A, b, Aeq, beq)
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
1.5000 1.5000
For more info, please lookup
help fmincon
FMINCON finds a constrained minimum of a function of several variables.
FMINCON attempts to solve problems of the form:
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 (bounds)
FMINCON implements four different algorithms: interior point, SQP,
active set, and trust region reflective. Choose one via the option
Algorithm: for instance, to choose SQP, set OPTIONS =
optimoptions('fmincon','Algorithm','sqp'), and then pass OPTIONS to
FMINCON.
X = FMINCON(FUN,X0,A,B) starts at X0 and finds a minimum X to the
function FUN, subject to the linear inequalities A*X <= B. FUN accepts
input X and returns a scalar function value F evaluated at X. X0 may be
a scalar, vector, or matrix.
X = FMINCON(FUN,X0,A,B,Aeq,Beq) minimizes FUN subject to the linear
equalities Aeq*X = Beq as well as A*X <= B. (Set A=[] and B=[] if no
inequalities exist.)
X = FMINCON(FUN,X0,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.
X = FMINCON(FUN,X0,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. FMINCON minimizes FUN such
that C(X) <= 0 and Ceq(X) = 0. (Set LB = [] and/or UB = [] if no bounds
exist.)
X = FMINCON(FUN,X0,A,B,Aeq,Beq,LB,UB,NONLCON,OPTIONS) minimizes with
the default optimization parameters replaced by values in OPTIONS, an
argument created with the OPTIMOPTIONS function. See OPTIMOPTIONS for
details. For a list of options accepted by FMINCON refer to the
documentation.
X = FMINCON(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a
structure with the function FUN in PROBLEM.objective, the start point
in PROBLEM.x0, 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 nonlinear constraint function in PROBLEM.nonlcon, the
options structure in PROBLEM.options, and solver name 'fmincon' in
PROBLEM.solver. Use this syntax to solve at the command line a problem
exported from OPTIMTOOL.
[X,FVAL] = FMINCON(FUN,X0,...) returns the value of the objective
function FUN at the solution X.
[X,FVAL,EXITFLAG] = FMINCON(FUN,X0,...) returns an EXITFLAG that
describes the exit condition. Possible values of EXITFLAG and the
corresponding exit conditions are listed below. See the documentation
for a complete description.
All algorithms:
1 First order optimality conditions satisfied.
0 Too many function evaluations or iterations.
-1 Stopped by output/plot function.
-2 No feasible point found.
Trust-region-reflective, interior-point, and sqp:
2 Change in X too small.
Trust-region-reflective:
3 Change in objective function too small.
Active-set only:
4 Computed search direction too small.
5 Predicted change in objective function too small.
Interior-point and sqp:
-3 Problem seems unbounded.
[X,FVAL,EXITFLAG,OUTPUT] = FMINCON(FUN,X0,...) returns a structure
OUTPUT with information such as total number of iterations, and final
objective function value. See the documentation for a complete list.
[X,FVAL,EXITFLAG,OUTPUT,LAMBDA] = FMINCON(FUN,X0,...) returns the
Lagrange multipliers at the solution X: LAMBDA.lower for LB,
LAMBDA.upper for UB, LAMBDA.ineqlin is for the linear inequalities,
LAMBDA.eqlin is for the linear equalities, LAMBDA.ineqnonlin is for the
nonlinear inequalities, and LAMBDA.eqnonlin is for the nonlinear
equalities.
[X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD] = FMINCON(FUN,X0,...) returns the
value of the gradient of FUN at the solution X.
[X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = FMINCON(FUN,X0,...)
returns the value of the exact or approximate Hessian of the Lagrangian
at X.
Examples
FUN can be specified using @:
X = fmincon(@humps,...)
In this case, F = humps(X) returns the scalar function value F of
the HUMPS function evaluated at X.
FUN can also be an anonymous function:
X = fmincon(@(x) 3*sin(x(1))+exp(x(2)),[1;1],[],[],[],[],[0 0])
returns X = [0;0].
If FUN or NONLCON are parameterized, you can use anonymous functions to
capture the problem-dependent parameters. Suppose you want to minimize
the objective given in the function myfun, subject to the nonlinear
constraint mycon, where these two functions are parameterized by their
second argument a1 and a2, respectively. Here myfun and mycon are
MATLAB file functions such as
function f = myfun(x,a1)
f = x(1)^2 + a1*x(2)^2;
function [c,ceq] = mycon(x,a2)
c = a2/x(1) - x(2);
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 myfun and
mycon with two arguments. Finally, pass these anonymous functions to
FMINCON:
a1 = 2; a2 = 1.5; % define parameters first
options = optimoptions('fmincon','Algorithm','interior-point'); % run interior-point algorithm
x = fmincon(@(x) myfun(x,a1),[1;2],[],[],[],[],[],[],@(x) mycon(x,a2),options)
See also OPTIMOPTIONS, OPTIMTOOL, FMINUNC, FMINBND, FMINSEARCH, @, FUNCTION_HANDLE.
Documentation for fmincon
doc fmincon
