Quadratic Programming
Solves quadratic programming problems with linear and box constraints.
quadprog(C, d, A = NULL, b = NULL,
Aeq = NULL, beq = NULL, lb = NULL, ub = NULL)C |
symmetric matrix, representing the quadratic term. |
d |
vector, representing the linear term. |
A |
matrix, represents the linear constraint coefficients. |
b |
vector, constant vector in the constraints. |
Aeq |
matrix, linear equality constraint coefficients. |
beq |
vector, constant equality constraint vector. |
lb |
elementwise lower bounds. |
ub |
elementwise upper bounds. |
Finds a minimum for the quadratic programming problem specified as:
min 1/2 x'Cx + d'x
such that the following constraints are satisfied:
A x <= b
Aeq x = beq
lb <= x <= ub
The matrix should be symmetric and positive definite, in which case the solution is unique, indicated when the exit flag is 1.
For more information, see ?solve.QP.
Returns a list with components
xmin |
minimum solution, subject to all bounds and constraints. |
fval |
value of the target expression at the arg minimum. |
eflag |
exit flag. |
This function is wrapping the active set quadratic solver in the
quadprog package: quadprog::solve.QP, combined with
a more MATLAB-like API interface.
Nocedal, J., and St. J. Wright (2006). Numerical Optimization. Second Edition, Springer Series in Operations Research, New York.
lsqlincon, quadprog::solve.QP
## Example in ?solve.QP # Assume we want to minimize: 1/2 x^T x - (0 5 0) %*% x # under the constraints: A x <= b # with b = (8,-2, 0) # and ( 4 3 0) # A = (-2 -1 0) # ( 0 2,-1) # and possibly equality constraint 3x1 + 2x2 + x3 = 1 # or upper bound c(1.5, 1.5, 1.5). C <- diag(1, 3); d <- -c(0, 5, 0) A <- matrix(c(4,3,0, -2,-1,0, 0,2,-1), 3, 3, byrow=TRUE) b <- c(8, -2, 0) quadprog(C, d, A, b) # $xmin # [1] 0.4761905 1.0476190 2.0952381 # $fval # [1] -2.380952 # $eflag # [1] 1 Aeq <- c(3, 2, 1); beq <- 1 quadprog(C, d, A, b, Aeq, beq) # $xmin # [1] 1.4 -0.8 -1.6 # $fval # [1] 6.58 # $eflag # [1] 1 quadprog(C, d, A, b, lb = 0, ub = 1.5) # $xmin # [1] 0.625 0.750 1.500 # $fval # [1] -2.148438 # $eflag # [1] 1 ## Example help(quadprog) C <- matrix(c(1, -1, -1, 2), 2, 2) d <- c(-2, -6) A <- matrix(c(1,1, -1,2, 2,1), 3, 2, byrow=TRUE) b <- c(2, 2, 3) lb <- c(0, 0) quadprog(C, d, A, b, lb=lb) # $xmin # [1] 0.6666667 1.3333333 # $fval # [1] -8.222222 # $eflag # [1] 1
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