SigmoidalPrograming

SigmoidalProgramming is a Julia package for solving sigmoidal programming problems. It solves problems using a branch and bound method described here, and solves the convex subproblems using a cutting plane method.

Usage examples can be found in the examples directory.

You can install the package from the Julia prompt

Pkg.add("SigmoidalProgramming")

The LinearSP problem class can be used to solve any sigmoidal programming problem with linear equality and inequality constraints, of the form

    minimize       sum_i f_i(x_i)
    subject to     A x <= b
                   C x == d
                   l <= x <= u

with variable x, where f_i is a sigmoidal function for every i. A sigmoidal function is a function f(x) that, for some number z, is convex in x for x <= z and is concave in x for x >= z. (Convex functions are sigmoidal with z = float("inf"), and concave functions are sigmoidal with z = -float("inf").)

The functions f are specified with three lists:

1. fs, a list of functions such that fs[i](x) computes the value of f_i at x.
2. dfs, a list of functions such that dfs[i](x) computes the derivative of f_i at x.
3. z, a list of numbers such that f_i is convex in its argument x for x <= z and is concave in x for x >= z.

Let's see how to use it to solve a random sigmoidal program with only inequality constraints. We'll let f_i = x -> 1/(1 + exp(-x)) be the logistic function for every i. The logistic function and its derivative logistic_prime are defined in the file functions.jl.

using SigmoidalProgramming

# generate problem data
srand(4)
nvar = 200
nineqconstr = 20
l = -rand(nvar)
u = rand(nvar)
A = rand(nineqconstr, nvar)
b = rand(nineqconstr)
z = zeros(nvar)
fs = fill(logistic, nvar)
dfs = fill(logistic_prime, nvar)
problem = LinearSP(fs, dfs, z, A, b)

# branch and bound to solve the problem
# pq is a priority queue of the branch and bound nodes at the leaves of the tree
# bestnodes is a list of the best branch and bound nodes found, in the order they were found
pq, bestnodes, lbs, ubs = solve_sp(l, u, problem)

# the best node found yet is the top node on the priority queue
node = dequeue!(pq)
# println("best node has node.ub = $(node.ub) and solution $(node.x)")

# lbs and ubs record the upper and lower bounds on the optimal value
# found at each iteration
println("lbs: ",lbs)
println("ubs: ",ubs)

A problem with equality constraints C x = d as well can be solved using problem = LinearSP(fs, dfs, z, A, b, C, d)

A few simple examples of usage can be found in the examples directory.

The accuracy of the solution can be controlled with the parameters

• maxiters, the maximum number of branch and bound iterations (default: 100)
• TOL, the desired gap between the upper and lower bound on the optimal value (default: 1e-2)

The solver terminates as soon as either maxiters branch and bound nodes have been split, or the gap between the upper and lower bound on the optimal value has been proved to be less than TOL.

The call signature for solve_sp is as follows:

solve_sp(l, u, problem::SigmoidalProgram; 
                  TOL = 1e-2, maxiters = 100, verbose = false)