# 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

Missing docstring.

Missing docstring for SIPProblem. Check Documenter's build log for details.

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Missing docstring for SIPResult. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SIPRes. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SIPResRev. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SIPHybrid. Check Documenter's build log for details.

Missing docstring.

Missing docstring for get_sip_optimizer. Check Documenter's build log for details.

Missing docstring.

Missing docstring for build_model. Check Documenter's build log for details.

Missing docstring.

Missing docstring for sip_llp!. Check Documenter's build log for details.

Missing docstring.

Missing docstring for sip_bnd!. Check Documenter's build log for details.

Missing docstring.

Missing docstring for sip_res!. Check Documenter's build log for details.

Missing docstring.

Missing docstring for sip_solve. Check Documenter's build log for details.

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.