Documentation 
Quadratic programming
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,...)
Finds a minimum for a problem specified by
$$\underset{x}{\mathrm{min}}\frac{1}{2}{x}^{T}Hx+{f}^{T}x\text{suchthat}\{\begin{array}{c}A\cdot x\le b,\\ Aeq\cdot x=beq,\\ lb\le x\le ub.\end{array}$$
H, A, and Aeq are matrices, and f, b, beq, lb, ub, and x are vectors.
f, lb, and ub can be passed as vectors or matrices; see Matrix Arguments.
x = quadprog(H,f) returns a vector x that minimizes 1/2*x'*H*x + f'*x. H must be positive definite for the problem to have a finite minimum.
x = quadprog(H,f,A,b) minimizes 1/2*x'*H*x + f'*x subject to the restrictions A*x ≤ b. A is a matrix of doubles, and b is a vector of doubles.
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) solves the preceding problem starting from the vector x0. If no bounds exist, set lb = [] and ub = []. Some quadprog algorithms ignore x0, see Input Arguments.
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.
[x,fval] = quadprog(H,f,...) returns the value of the objective function at x:
fval = 0.5*x'*H*x + f'*x
[x,fval,exitflag] = quadprog(H,f,...) exitflag, a scalar that describes the exit condition of quadprog.
[x,fval,exitflag,output] = quadprog(H,f,...) output, a structure that contains information about the optimization.
[x,fval,exitflag,output,lambda] = quadprog(H,f,...) lambda, a structure whose fields contain the Lagrange multipliers at the solution x.
H 
Symmetric matrix of doubles. Represents the quadratic in the expression 1/2*x'*H*x + f'*x.  
f 
Vector of doubles. Represents the linear term in the expression 1/2*x'*H*x + f'*x.  
A 
Matrix of doubles. Represents the linear coefficients in the constraints A*x ≤ b.  
b 
Vector of doubles. Represents the constant vector in the constraints A*x ≤ b.  
Aeq 
Matrix of doubles. Represents the linear coefficients in the constraints Aeq*x = beq.  
beq 
Vector of doubles. Represents the constant vector in the constraints Aeq*x = beq.  
lb 
Vector of doubles. Represents the lower bounds elementwise in lb ≤ x ≤ ub.  
ub 
Vector of doubles. Represents the upper bounds elementwise in lb ≤ x ≤ ub.  
x0 
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 interiorpointconvex algorithm, and for the trustregionreflective algorithm with equality constraints.  
options 
Options created using optimoptions or the Optimization app. All Algorithms
All Algorithms Except activeset
trustregionreflective Algorithm Only
interiorpointconvex Algorithm Only
 
problem 
Structure encapsulating the quadprog inputs and options:

x 
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.  
fval 
Value of 1/2*x'*H*x + f'*x at the solution x, a double.  
exitflag 
Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated:
 
output 
Structure containing information about the optimization. The fields are:
 
lambda 
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
$$f(x)=\frac{1}{2}{x}_{1}^{2}+{x}_{2}^{2}{x}_{1}{x}_{2}2{x}_{1}6{x}_{2},$$
subject to
x_{1} + x_{2} ≤
2
–x_{1} +
2x_{2} ≤ 2
2x_{1} + x_{2} ≤
3
0 ≤ x_{1},
0 ≤ x_{2}.
In matrix notation this is
$$f(x)=\frac{1}{2}{x}^{T}Hx+{f}^{T}x,$$
where
$$H=\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right],\text{}f=\left[\begin{array}{c}2\\ 6\end{array}\right],\text{}x=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right].$$
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 'activeset' algorithm with no display:
opts = optimoptions('quadprog','Algorithm','activeset','Display','off');
Call quadprog:
[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 'interiorpointconvex' 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 'interiorpointconvex' algorithm and iterative display:
opts = optimoptions('quadprog',... 'Algorithm','interiorpointconvex','Display','iter');
Run the quadprog solver and observe the iterations:
[x fval eflag output lambda] = quadprog(H,f,A,b,[],[],[],[],[],opts); Firstorder 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 9.465e002 9.465e002 2 2.639877e+001 0.000e+000 3.914e005 3.914e005 3 2.639881e+001 0.000e+000 3.069e015 6.883e015 Minimum found that satisfies the constraints. Optimization completed because the objective function is nondecreasing 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 'interiorpointconvex' 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.
[1] 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.
[2] Gill, P. E., W. Murray, and M. H. Wright, Practical Optimization, Academic Press, London, UK, 1981.
[3] Gould, N. and P. L. Toint. "Preprocessing for quadratic programming." Math. Programming, Series B, Vol. 100, pp. 95–132, 2004.