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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

minx12xTHx+fTx such that {Axb,Aeqx=beq,lbxub.

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.

Input Arguments


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 the trust-region-reflective algorithm when there are only bound constraints.

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.

Some options are absent from the optimoptions display. These options are listed in italics. For details, see View Options.

All Algorithms


Choose the algorithm:

  • 'interior-point-convex' (default)

  • 'trust-region-reflective'

The 'trust-region-reflective' algorithm handles problems with only bounds, or only linear equality constraints, but not both. The 'interior-point-convex' algorithm handles only convex problems. For details, see Choosing the Algorithm.


Display diagnostic information about the function to be minimized or solved. The choices are 'on' or 'off' (default).


Level of display (see Iterative Display):

  • 'off' or 'none' displays no output.

  • 'final' displays just the final output (default).

The 'interior-point-convex' algorithm allows additional values:

  • 'iter' gives iterative display.

  • 'iter-detailed' gives iterative display with a detailed exit message.

  • 'final-detailed' displays just the final output, with a detailed exit message.


Maximum number of iterations allowed, a positive integer.

  • For a 'trust-region-reflective' equality-constrained problem, the default value is 2*(numberOfVariables - numberOfEqualities).

  • For all other algorithms and problems, the default value is 200.


Termination tolerance on the first-order optimality, a positive scalar.

  • For a 'trust-region-reflective' equality-constrained problem, the default value is 1e-6.

  • For a 'trust-region-reflective' bound-constrained problem, the default value is 100*eps, about 2.2204e-14.

  • For 'interior-point-convex', the default value is 1e-8.

See Tolerances and Stopping Criteria.


Termination tolerance on x, a positive scalar.

  • For 'trust-region-reflective', the default value is 100*eps, about 2.2204e-14.

  • For 'interior-point-convex', the default value is 1e-8.

trust-region-reflective Algorithm Only


Termination tolerance on the function value, a positive scalar. The default is depends on the problem type: bound constrained problems use 100*eps, linear equality constrained problems use 1e-6. See Tolerances and Stopping Criteria.


Function handle for a Hessian multiply function. For large-scale structured problems, this function computes the Hessian matrix product H*Y without actually forming H. The function has the form

W = hmfun(Hinfo,Y)

where Hinfo and possibly some additional parameters contain the matrices used to compute H*Y.

See Quadratic Minimization with Dense, Structured Hessian for an example that uses this option.


Maximum number of PCG (preconditioned conjugate gradient) iterations, a positive scalar. The default is max(1,floor(numberOfVariables/2)). For more information, see Preconditioned Conjugate Gradient Method.


Upper bandwidth of the preconditioner for PCG, a nonnegative integer. By default, quadprog uses diagonal preconditioning (upper bandwidth 0). For some problems, increasing the bandwidth reduces the number of PCG iterations. Setting PrecondBandWidth to Inf uses a direct factorization (Cholesky) rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step toward the solution.


Determines how the iteration step is calculated. The default, 'cg', takes a faster but less accurate step than 'factorization'. See trust-region-reflective quadprog Algorithm.


Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1.


Typical x values. The number of elements in TypicalX equals the number of elements in x0, the starting point. The default value is ones(numberofvariables,1). quadprog uses TypicalX internally for scaling. TypicalX has an effect only when x has unbounded components, and when a TypicalX value for an unbounded component exceeds 1.

interior-point-convex Algorithm Only


Tolerance on the constraint violation, a positive scalar. The default is 1e-8.


Structure encapsulating the quadprog inputs and options:


Symmetric matrix in 1/2*x'*H*x


Vector in linear term f'*x


Matrix in linear inequality constraints Aineq*x  bineq


Vector in linear inequality constraints Aineq*x  bineq


Matrix in linear equality constraints Aeq*x = beq


Vector in linear equality constraints Aeq*x = beq
lbVector of lower bounds
ubVector of upper bounds


Initial point for x




Options created using optimoptions or the Optimization app

Output Arguments


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:

All Algorithms


Function converged to the solution x.


Number of iterations exceeded options.MaxIterations.


Problem is infeasible.


Problem is unbounded.

interior-point-convex Algorithm


Nonconvex problem detected.

trust-region-reflective Algorithm


Change in the objective function value was smaller than options.FunctionTolerance.


Current search direction was not a direction of descent. No further progress could be made.


Structure containing information about the optimization. The fields are:


Number of iterations taken


Optimization algorithm used


Total number of PCG iterations (trust-region-reflective algorithm only)


Maximum of constraint functions


Measure of first-order optimality


Exit message


Structure containing the Lagrange multipliers at the solution x (separated by constraint type). The fields are:


Lower bounds lb


Upper bounds ub


Linear inequalities


Linear equalities

For details, see Lagrange Multiplier Structures.


Solve a simple quadratic programming problem: find values of x that minimize


subject to

x1 + x2 ≤ 2
x1 + 2x2 ≤ 2
2x1 + x2 ≤ 3
0 ≤ x1, 0 ≤ x2.

In matrix notation this is



H=[1112],  f=[26],  x=[x1x2].

  1. 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);
  2. Set the options to use the 'interior-point-convex' algorithm with no display:

    options = optimoptions('quadprog',...
  3. Call quadprog:

    [x,fval,exitflag,output,lambda] = ...
  4. Examine the final point, function value, and exit flag:

    x =
    fval =
    exitflag =
  5. 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:

    ans =

Use the 'interior-point-convex' algorithm to solve a sparse quadratic program.

  1. Generate a sparse symmetric matrix for the quadratic form:

    v = sparse([1,-.25,0,0,0,0,0,-.25]);
    H = gallery('circul',v);
  2. Include the linear term for the problem:

    f = -4:3;
  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;
  4. Set options to use the 'interior-point-convex' algorithm and iterative display:

    opts = optimoptions('quadprog',...
  5. 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.
  6. Examine the solution:

    fval =
    eflag =

    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.

More About

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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.


The 'trust-region-reflective' algorithm is a subspace trust-region method based on the interior-reflective Newton method described in [1]. 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.


[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.

Introduced before R2006a

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