(253c) Online Generation Via Offline Selection of Strong Linear Cuts from a Semidefinite Programming Relaxation

Misener, R., Imperial College
Baltean-Lugojan, R., Imperial College London
Bonami, P., IBM
Deterministic global optimization is highly relevant solving quadratically-constrained quadratic optimisation (QCQP) problems that arise in a variety of engineering applications [1]. In a branch-and-cut framework, cutting planes are known to significantly expedite the solution for certain classes of QCQP problems [2-4].

Convex, particularly semidefinite programming (SDP), relaxations for non-convex continuous quadratic optimization can provide tighter bounds than the reformulation-linearization technique [5]. But using SDP relaxations directly in branch and cut is impeded by a lack of warm starting and inefficiency when combined with linear cut classes, e.g. the McCormick bilinear envelopes [6] or the reformulation-linearization technique [7]. State-of-the-art deterministic global optimization solvers use linear relaxations [4, 8-12], so SDP technology is effectively unavailable.

Previous work in developing SDP-based cutting planes has shown that these cuts can be very effective for certain classes of QCQP [13, 14]. But it’s still unclear when to generate these relatively expensive cuts, i.e. when to solve an SDP problem with the goal of generating a cutting plane. We seek an inexpensive oracle that will tell us when it is relevant to generate a cut and over which variables.

We present a general framework based on machine learning for a strong linear outer-approximation that can retain most tightness of such SDP relaxations, in the form of few strong low-dimensional linear cuts selected offline. The cut selection complexity is taken offline by using a neural network estimator (trained before installing solver software) as a selection device for the strongest cuts. Once the neural network estimator selects the appropriate subsets of variables, we generate hyperplanes cutting off the current objective based on negative eigenvalues [15, 16].

We present results of our method on several classes of non-convex application problems from the engineering literature. We also test the strategy on generic QCQP and integrate into the global optimization solver ANTIGONE [17].


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