QPmin (version 0.5-1)

QPgen.internal.concave: QPgen.internal.concave

Description

Generates a separable strictly concave quadratic problem of the form $$ \min_x \frac{1}{2} x^T G x + x^T g $$ $$ Ax \geq b $$

Usage

QPgen.internal.concave(m, thetas, alphas, betas, L)

Arguments

m

Integer parameter controlling the number of variables (2m) and constraints (3m) for the generated problem.

thetas

m binary {0, 1} parameters.

alphas

m parameters taking on either of the two values {1.5, 2}

betas

m parameters taking on either of the two value {1.5, 2}. Each entry of betas must be different from each entry of alphas.

L

A positive parameter

Value

G

The quadratic component of the objective function.

g

The linear component of the objective function

A

The constraints coefficient matrix. This matrix has 3m rows and 2m columns.

b

The vector with the lower bounds on the constraints.

opt

An approximate expected value at the optimum solutions.

globals

A list containing all of the global solutions to the problem.

Details

Denoting m0 the number of thetas == 0, the generated problem has a unique global minimum and 3^m0 local ones. All of the local minima have distinct function values. While the constraints are guaranteed to be indipendent at each solution, they are not guaranteed independent everywhere, for this reason too large problems tend to not be solvable with QPmin, which relies on a basic version of the simplex method for linear programming to find an initial feasible point. A large enough problem will generally halt the execution due to linear dependence in the basis constraints.

References

``A new technique for generating quadratic programming test problems,'' Calamai P.H., L.N. Vicente, and J.J. Judice, Mathematical Programming 61 (1993), pp. 215-231.

Examples

# NOT RUN {
m <- 3
thetas <- round(runif(m))
draws <- runif(m)
belowhalf <- draws < 0.5 
alphas <- betas <- c()
alphas[belowhalf] <- 1.5
alphas[!belowhalf] <- 2
betas[belowhalf] <- 2
betas[!belowhalf] <- 1.5
L <- ceiling(log(m))
QPgen.internal.concave(m, thetas, alphas, betas, L)
# }