x = quadprog(H,f)
x = quadprog(H,f,A,b)
x = quadprog(H,f,A,b,Aeq,beq)
x = quadprog(H,f,A,b,Aeq,beq,lb,ub)
x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0)
x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options)
x = quadprog(problem)
[x,fval] = quadprog(H,f,...)
[x,fval,exitflag] = quadprog(H,f,...)
[x,fval,exitflag,output] = quadprog(H,f,...)
[x,fval,exitflag,output,lambda] = quadprog(H,f,...)
x = quadprog(H,f,A,b,Aeq,beq) solves the preceding problem subject to the additional restrictions Aeq*x = beq. Aeq is a matrix of doubles, and beq is a vector of doubles. If no inequalities exist, set A =  and b = .
x = quadprog(H,f,A,b,Aeq,beq,lb,ub) solves the preceding problem subject to the additional restrictions lb ≤ x ≤ ub. lb and ub are vectors of doubles, and the restrictions hold for each x component. If no equalities exist, set Aeq =  and beq = .
Note: If the specified input bounds for a problem are inconsistent, the output x is x0 and the output fval is .
quadprog resets components of x0 that violate the bounds lb ≤ x ≤ ub to the interior of the box defined by the bounds. quadprog does not change components that respect the bounds.
x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options) solves the preceding problem using the optimization options specified in options. Use optimoptions to create options. If you do not want to give an initial point, set x0 = .
x = quadprog(problem) returns the minimum for problem, where problem is a structure described in Input Arguments. Create problem by exporting a problem using the Optimization app; see Exporting Your Work.
fval = 0.5*x'*H*x + f'*x
Symmetric matrix of doubles. Represents the quadratic in the expression 1/2*x'*H*x + f'*x.
Vector of doubles. Represents the linear term in the expression 1/2*x'*H*x + f'*x.
Matrix of doubles. Represents the linear coefficients in the constraints A*x ≤ b.
Vector of doubles. Represents the constant vector in the constraints A*x ≤ b.
Matrix of doubles. Represents the linear coefficients in the constraints Aeq*x = beq.
Vector of doubles. Represents the constant vector in the constraints Aeq*x = beq.
Vector of doubles. Represents the lower bounds elementwise in lb ≤ x ≤ ub.
Vector of doubles. Represents the upper bounds elementwise in lb ≤ x ≤ ub.
Vector of doubles. Optional. The initial point for some quadprog algorithms:
If you do not give x0, quadprog sets all components of x0 to a point in the interior of the box defined by the bounds. quadprog ignores x0 for the interior-point-convex algorithm, and for the trust-region-reflective algorithm with equality constraints.
Options created using optimoptions or the Optimization app.
All Algorithms Except active-set
trust-region-reflective Algorithm Only
interior-point-convex Algorithm Only
Structure encapsulating the quadprog inputs and options:
Vector that minimizes 1/2*x'*H*x + f'*x subject to all bounds and linear constraints. x can be a local minimum for nonconvex problems. For convex problems, x is a global minimum. For more information, see Local vs. Global Optima.
Value of 1/2*x'*H*x + f'*x at the solution x, a double.
Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated:
Structure containing information about the optimization. The fields are:
Structure containing the Lagrange multipliers at the solution x (separated by constraint type). The fields are:
For details, see Lagrange Multiplier Structures.
Solve a simple quadratic programming problem: find values of x that minimize
x1 + x2 ≤
–x1 + 2x2 ≤ 2
2x1 + x2 ≤ 3
0 ≤ x1, 0 ≤ x2.
In matrix notation this is
Enter the coefficient matrices:
H = [1 -1; -1 2]; f = [-2; -6]; A = [1 1; -1 2; 2 1]; b = [2; 2; 3]; lb = zeros(2,1);
Set the options to use the active-set algorithm with no display:
opts = optimoptions('quadprog','Algorithm','active-set','Display','off');
[x,fval,exitflag,output,lambda] = ... quadprog(H,f,A,b,,,lb,,,opts);
Examine the final point, function value, and exit flag:
x,fval,exitflag x = 0.6667 1.3333 fval = -8.2222 exitflag = 1
An exit flag of 1 means the result is a local minimum. Because H is a positive definite matrix, this problem is convex, so the minimum is a global minimum. You can see H is positive definite by noting all its eigenvalues are positive:
eig(H) ans = 0.3820 2.6180
Use the interior-point-convex algorithm to solve a sparse quadratic program.
Generate a sparse symmetric matrix for the quadratic form:
v = sparse([1,-.25,0,0,0,0,0,-.25]); H = gallery('circul',v);
Include the linear term for the problem:
f = -4:3;
Include the constraint that the sum of the terms in the solution x must be less than -2:
A = ones(1,8);b = -2;
Set options to use the interior-point-convex algorithm and iterative display:
opts = optimoptions('quadprog',... 'Algorithm','interior-point-convex','Display','iter');
Run the quadprog solver and observe the iterations:
[x fval eflag output lambda] = quadprog(H,f,A,b,,,,,,opts); First-order Total relative Iter f(x) Feasibility optimality error 0 -2.000000e+000 1.000e+001 4.500e+000 1.200e+001 1 -2.630486e+001 0.000e+000 9.465e-002 9.465e-002 2 -2.639877e+001 0.000e+000 3.914e-005 3.914e-005 3 -2.639881e+001 0.000e+000 3.069e-015 6.883e-015 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.
Examine the solution:
fval,eflag fval = -26.3988 eflag = 1
For the interior-point-convex algorithm, an exit flag of 1 means the result is a global minimum.
You can use the Optimization app for quadratic programming. Enter optimtool at the MATLAB® command line, and choose the quadprog - Quadratic programming solver. For more information, see Optimization App.
The trust-region-reflective algorithm is a subspace trust-region method based on the interior-reflective Newton method described in . Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). For more information, see trust-region-reflective quadprog Algorithm.
quadprog uses an active set method, which is also a projection method, similar to that described in . It finds an initial feasible solution by first solving a linear programming problem. For more information, see active-set quadprog Algorithm.
The interior-point-convex algorithm attempts to follow a path that is strictly inside the constraints. It uses a presolve module to remove redundancies, and to simplify the problem by solving for components that are straightforward. For more information, see interior-point-convex quadprog Algorithm.
 Coleman, T.F. and Y. Li, "A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables," SIAM Journal on Optimization, Vol. 6, Number 4, pp. 1040–1058, 1996.
 Gill, P. E., W. Murray, and M. H. Wright, Practical Optimization, Academic Press, London, UK, 1981.
 Gould, N. and P. L. Toint. "Preprocessing for quadratic programming." Math. Programming, Series B, Vol. 100, pp. 95–132, 2004.