Solving Semi-Infinite Programming

Matthew Wilhelm Department of Chemical and Biomolecular Engineering, University of Connecticut

Using EAGO to solve a SIP

Semi-infinite programming remains an active area of research. In general, the solution of semi-infinite programs with nonconvex semi-infinite constraints of the below form are extremely challenging:

EAGO implements three different algorithm detailed in [1,2] to determine a globally optimal solution to problems of the above form. This accomplished using the sip_solve function which returns the optimal value, the solution, and a boolean feasibility value. To illustrate the functions use, a simple example is presented here which solves the below problem:

using EAGO, JuMP

# Define semi-infinite program
f(x) = (1/3)*x[1]^2 + x[2]^2 + x[1]/2
gSIP(x,p) = (1.0 - (x[1]^2)*(p[1]^2))^2 - x[1]*p[1]^2 - x[2]^2 + x[2]

x_l = Float64[-1000.0, -1000.0]
x_u = Float64[1000.0, 1000.0]
p_l = Float64[0.0]
p_u = Float64[1.0]

sip_result = sip_solve(SIPRes(), x_l, x_u, p_l, p_u, f, Any[gSIP], abs_tolerance = 1E-3)

println("The global minimum of the semi-infinite program is between: $(sip_result.lower_bound) and $(sip_result.upper_bound).")
println("The global minimum is attained at: x = $(sip_result.xsol).")
println("Is the problem feasible? $(sip_result.feasibility).")

Semi-infinite solver

1. Mitsos A (2009). Global optimization of semi-infinite programs via restriction of the right-hand side. Optimization, 60(10-11):1291-1308.
2. Djelassi, Hatim, and Alexander Mitsos. A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs. Journal of Global Optimization, 68.2 (2017): 227-253 should be used.