On strong valid inequalities for a class of mixed-integer nonlinear sets with box constraints
In this paper, we investigate the mixed-integer nonlinear set with box constraints $X = \{(w,x)\in R\times Z^n:w\leq f(a^Tx),0\leq x\leq \mu\}$, where $f$ is a univariate concave function, $a\in R^n$, and $\mu\in Z^n_{++}$. This set arises as a substructure in many mixed-integer nonlinear optimization models and encompasses, as special cases, several previously investigated mixed-integer sets, namely the submodular maximization set, the mixed-integer knapsack set, and the mixed-integer polyhedral conic set. We present the first comprehensive polyhedral study of conv($X$). In particular, we derive a class of seed inequalities for a two-dimensional restriction of $X$, obtained by fixing all but one of the $x$ variables to their bounds in $X$, and develop two lifting procedures to obtain strong valid inequalities for conv($X$). In the first lifting procedure, we derive a subadditive approximation for the exact lifting function of the seed inequalities, and lift all fixed variables in a single phase. In the second lifting procedure, we first lift variables fixed at their lower bounds before those at their upper bounds (and vice versa), using subadditive exact and approximation lifting functions, respectively. The derived single- and two-phase lifted inequalities are shown to be facet-defining for conv($X$) under mild conditions. Moreover, for the aforementioned special cases of conv($X$), we show that the proposed lifted inequalities can either unify existing strong valid inequalities or yield new facet-defining inequalities. Finally, extensive computational experiments on expected utility maximization and weapon-target assignment problems demonstrate that the proposed lifted inequalities can substantially strengthen the continuous relaxations and significantly improve the overall computational performance of branch-and-cut algorithms.