Global Solutions for Mixed-Integer Nonlinear Optimization

Deterministic global optimization of mixed-integer nonlinear programs (MINLP) is broadly applicable in diverse domains ranging from molecular biology to refinery operations to computational chemistry to synthesizing sustainable processes. MINLP is mathematically defined:

where C, B, I, and M represent the number of continuous variables, binary variables, integer variables, and constraints, respectively. Parameter vectors bmLO and bmUP bound the constraints. We assume that it is possible to infer finite bounds [ xiL, xiU ] on the variables participating in nonlinear terms fm and that the image of fm is finite on x. Typical expressions for f0(x, y, z) and fm(x, y, z) are:

where the powers psm, c are constant reals; cm, am, Qm, csm, cem, clm are constant coefficients; Sm, Em, Lm are the number of signomial, exponential, and logarithmic terms, respectively.

ANTIGONE (Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations) is a computational framework for globally solving mixed-integer nonlinear optimization problems. ANTIGONE is available through Princeton University and GAMS (beginning distribution 24.1).

The aggregate performance profiles [Dolan and Moré; PTOOLS] for the complete test suite diagrammed in the figures display a strong advantage for ANTIGONE on 2571 test cases across a wide array of problem classes:

TimePercent gap remaining

ANTIGONE is the fastest solver for ≈ 41% of the test cases and can address ≈ 60% of the test cases within the time limit. Additionally, ANTIGONE generally closes more of the optimality gap. The success of ANTIGONE is a direct result of finding and exploiting an array of special structure components within MINLP.