Function File: quadprog (H, f)

Function File: quadprog (H, f, A, b)

Function File: quadprog (H, f, A, b, Aeq, beq)

Function File: quadprog (H, f, A, b, Aeq, beq, lb, ub)

Function File: quadprog (H, f, A, b, Aeq, beq, lb, ub, x0)

Function File: quadprog (H, f, A, b, Aeq, beq, lb, ub, x0, options)

Function File: [x, fval, exitflag, output, lambda] = quadprog (…)

Solve the quadratic program

min 0.5 x'*H*x + x'*f
 x

subject to

A*x <= b,
Aeq*x = beq,
lb <= x <= ub.

The initial guess x0 and the constraint arguments (A and b, Aeq and beq, lb and ub) can be set to the empty matrix ([]) if not given. If the initial guess x0 is feasible the algorithm is faster.

options can be set with optimset, currently the only option is MaxIter, the maximum number of iterations (default: 200).

Returned values:

x

Position of minimum.

fval

Value at the minimum.

exitflag

Status of solution:

0

Maximum number of iterations reached.

-2

The problem is infeasible.

-3

The problem is not convex and unbounded

1

Global solution found.

4

Local solution found.

output

Structure with additional information, currently the only field is iterations, the number of used iterations.

lambda

Structure containing Lagrange multipliers corresponding to the constraints. For equality constraints, the sign of the multipliers is chosen to satisfy the equation

0.5 H * x + f + A' * lambda_inequ + Aeq' * lambda_equ = 0 .

If lower and upper bounds are equal, or so close to each other that they are considered equal by the algorithm, only one of these bounds is considered active when computing the solution, and a positive lambda will be placed only at this bound.

This function calls Octave’s __qp__ back-end algorithm internally.

Demonstration 1

  C = [0.9501    0.7620    0.6153    0.4057
      0.2311    0.4564    0.7919    0.9354
      0.6068    0.0185    0.9218    0.9169
      0.4859    0.8214    0.7382    0.4102
      0.8912    0.4447    0.1762    0.8936];
  %% Linear Inequality Constraints
  d = [0.0578; 0.3528; 0.8131; 0.0098; 0.1388];
  A =[0.2027    0.2721    0.7467    0.4659
      0.1987    0.1988    0.4450    0.4186
      0.6037    0.0152    0.9318    0.8462];
  b =[0.5251; 0.2026; 0.6721];
  %% Linear Equality Constraints
  Aeq = [3 5 7 9];
  beq = 4;
  %% Bound constraints
  lb = -0.1*ones(4,1);
  ub = ones(4,1);
  H = C' * C;
  f = -C' * d;
  [x, obj, flag, output, lambda]=quadprog (H, f, A, b, Aeq, beq, lb, ub)

x =

  -0.1000
  -0.1000
   0.1599
   0.4090

obj = -0.3194
flag = 1
output =

  scalar structure containing the fields:

    iterations = 3

lambda =

  scalar structure containing the fields:

    lower =

       0.0674
       0.2499
            0
            0

    upper =

       0
       0
       0
       0

    eqlin = -0.016539
    ineqlin =

            0
       0.4982
            0

Package: optim