Abstract
We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the equality constraints. We show that our relaxations are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a well-known semidefinite program relaxation. The new relaxations are implemented in the global optimization solver BARON, and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that, for our test problems, these relaxations lead to a significant improvement in the performance of BARON.
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A Barrier coordinate minimization algorithm used to solve the nonsmooth regularized separation problem
A Barrier coordinate minimization algorithm used to solve the nonsmooth regularized separation problem
In this appendix, we briefly describe the algorithm proposed by Dong [5] to solve the nonsmooth regularized separation problem (39). We then describe our implementation of this algorithm. The algorithm operates on the following penalized log-det problem
where \(r_i(d_i) = \beta _i d_i\) for \(d_i > 0\), \(r_i(d_i) = \eta _i d_i\) for \(d_i \le 0\), \(\beta _i = \eta _i + \lambda , \; \forall i \in [n]\), and \(\sigma >0\). Each iteration of this algorithm involves the update of a feasible vector \({\bar{d}}\) and an inverse matrix \(V:= {\left[ Q + \text {diag}({\bar{d}}) + \alpha A^T A \right] }^{-1}\). Based on the optimality condition for (62), this algorithm performs coordinate minimization by choosing an index i determined as:
where \(\partial h({\bar{d}}; \sigma )\) is the subdifferential of \(h({\bar{d}}; \sigma )\). This choice of i leads to a one-dimensional minimization problem similar to (46) but involving \(h({\bar{d}} + \varDelta d_i e_i; \sigma )\). This problem can be solved analytically to obtain the following formula for \(\varDelta d_i^*\) (see Section 4 in [5] for details):
After calculating \(\varDelta d_i^*\) according to (64), \({\bar{d}}\) and V are updated using (50) and (51), respectively. Once (62) has been solved within a given precision, the penalty parameter \(\sigma \) is adjusted through a rule similar to (53):
where \(s({\bar{d}})\) is used as a measure of optimality. The relative improvement in the objective function of (39) is checked every \(\omega _{\text {check}} n\) iterations, and the algorithm terminates if this relative improvement is smaller than \(\epsilon _{\text {check}}\).
The entire procedure is summarized in Algorithm 3. In our implementation of this algorithm, we use an initial perturbation \({\hat{d}} = -1.5 \lambda _{\text {min}} (Q + \alpha A^T A) \mathbb {1}\). We set the following parameters by using the values recommended in [5]: \(\lambda = \sum _{i = 1}^{n} \eta _i\), \(\sigma _{\text {min}} = 10^{-5}\), \(\sigma _{\text {upd}} = 0.8\), \(\epsilon _{\text {upd}} = 0.03\). We use \(\text {MaxIter} = 500 n\), \(\omega _{\text {check}} = 10\) and \(\epsilon _{\text {check}} = 10^{-4}\). The initial value of \(\sigma _{\text {init}}\) is determined as:
where \( u_i \in \partial r_i({\hat{d}}_i)\), and \(\partial r_i({\hat{d}}_i)\) is the subdifferential of \(r_i({\hat{d}}_i)\).
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Nohra, C.J., Raghunathan, A.U. & Sahinidis, N.V. SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs. Math. Program. 196, 203–233 (2022). https://doi.org/10.1007/s10107-021-01680-9
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DOI: https://doi.org/10.1007/s10107-021-01680-9
Keywords
- Global optimization
- Mixed-integer quadratic programs
- Semidefinite programming relaxations
- Convex quadratic relaxations
- Quadratic cuts