# Find the minimum of a function

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Melika Eft on 24 Sep 2022
Commented: Sam Chak on 24 Sep 2022
How can I find the min of this function Minf0(x,y)=x^2+y^2 S.t x ≥ 2 , x+y=3

Sam Chak on 24 Sep 2022
Edited: Sam Chak on 24 Sep 2022
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
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
Sam Chak on 24 Sep 2022
@Torsten, thanks for the highlight.
@Melika Eft, please take note to learn that the solution is infeasible.