Improving the Approximated Projected Perspective Reformulation by dual information
Introduction
We study solution techniques for convex separable Mixed-Integer Non-Linear Programs (MINLP) with semi-continuous variables for which either assume the value or lie in the interval (). This can be expressed, introducing for , as We assume the functions to be closed convex, one time continuously differentiable and finite in the interval ; w.l.o.g. we also assume . In (P) we single out the linking constraints (2) that contain all the relationships linking the variables among them and with the other variables of the problem, except those (4) that “define” the semi-continuous nature of the . The reformulation technique developed in [7] require (2) to be empty; the extension developed in [1] allows to overcome this limitation, but potentially at the cost of a worse bound quality. The aim of this paper is to deal with constraint (2) in a cost-effective way. For our approach to work, (2) must have a compatible structure with that of (1); we initially assume equality constraints, with extensions discussed in Section 3. Because our approach hinges on availability of dual information for the continuous relaxation, we assume that the function in the “other variables ” and the “other constraints (3)” are convex, i.e., (P) is a convex MINLP. Actually, in many applications everything but (1) is linear. It will be sometimes expedient to refer to (3)–(4) as “”, and to as the set obtained by relaxing integrality constraints on and , if any.
Often, the most pressing issue in solving (P) is to derive tight lower bounds on its optimal value (P), which is typically done by solving its (convex) continuous relaxation (P) (we denote by (X) and (X), respectively, the optimal value and the continuous relaxation of any problem (X)). However, often (P) (P), making the solution approaches inefficient. The presence of semi-continuous variables has been exploited to propose reformulations (P) of (P) such that (P) (P) (P ) (P). This starts from considering (1) as , where if and , , and otherwise. The convex envelope of is known [4] to be – using the perspective function of – which yields the Perspective Reformulation of (P) As is convex, is convex for ; since if , can be extended by continuity assuming . Hence, (PR) is a convex MINLP if (P) is. Its continuous relaxation (PR)—the Perspective Relaxation of (P)—usually has (PR) (P), making (PR) a more convenient formulation [8], [9]. If is SOCP-representable then so is , hence the PR of a Mixed-Integer Second-Order Cone Program (MI-SOCP) is still a MI-SOCP. Thus, (PR) is not necessarily more complex to solve – and, sometimes, even less so [2]– than (P). Alternatively, one can consider a Semi-Infinite MINLP reformulation of (PR) where Perspective Cuts [4]– linear outer approximations of the epigraph of – are dynamically added. This is often the best approach [6], in particular for “general” (P) where no other structure is available. It is appropriate to remark that the (PR) approach also applies if the are vectors such that and , with a polytope; yet, here, as in [1], [7], each must be a single variable.
While (PR) provides a better bound, it is also usually more time consuming to solve than (P) because is “more complex” than . This trade-off is nontrivial, in particular if is “simple”. For instance, if is quadratic and everything else is linear, (P) is a Mixed-Integer Quadratic Program (MIQP) whereas (PR) is a MI-SOCP; hence, (P) – a QP – can be significantly cheaper to solve than (PR)—a SOCP. The Projected PR (PR) idea underpinning the approach studied here was indeed proposed in [7] for the quadratic case, and . It was then extended in [1] to a more general class of functions, and allowing . However, renders some of the arguments significantly more complex, hence for the sake of simplicity we will only present here the case where ; it will be plain to see that the arguments immediately extend to the more general one. The PR idea is to analyze as a function of only, i.e., projecting away : under appropriate assumptions, and if there are no linking constraints (2), this turns out to be a piecewise-convex functions with a “small” number of pieces, that can be characterized by just looking at the data of (P) (cf. (7)). Hence, (PR) can be reformulated in terms of piecewise-convex objective functions, which makes it easier to solve, especially when has some valuable structure (e.g., flow or knapsack) [7]. However, in several applications (2) are indeed present [1], [3], [4], [8], [10]. Furthermore, since the binary variables are removed from the formulation, branching has to be done “indirectly”, which rules out using off-the-shelf solvers. To overcome these two limitations, in [1] the Approximated PR (APR) reformulation has been proposed whereby the , after having been eliminated, are re-introduced in the formulation in order to encode the piecewise nature of . This is possible even if (2) are present, and it has the advantage that (APR) is still a MIQP if (P) is. However, (AP2R) (PR) may, and does, happen when linking constraints (2) are present, whence the “Approximate” moniker. This is still advantageous in some cases, but it may happen that the weaker bounds outweigh the faster solution time, making the approach not competitive with more straightforward implementations of the PR [1].
The aim of this paper is to improve the APR by presenting a simple and effective way to ensure that (AP2R) (PR) even if (2) are present, while keeping the shape of the formulation – and therefore, hopefully, the cost of (AP2R) – exactly the same. Since bound equivalence only holds at the root node of the B&C it is not obvious that the approach, despite the quicker solution times of (AP2R), is competitive. However, this is shown to be true in at least one relevant application, the Mean–Variance problem (with min buy-in and cardinality constraints) in portfolio optimization.
Section snippets
A quick overview of APR
We now quickly summarize the analysis in [1], albeit limited to the case , in order to prepare the ground for the new extension. We focus on the basic problem corresponding to one pair The analysis hinges on considering the (PR) of (P) rewritten as i.e., first minimizing with respect to , and then minimizing the resulting function
Improving APR using dual information
The idea is to reformulate (P) to include information about the linking constraints (2) in the objective function (1), so that it can be “processed” by the APR. This hinges on the availability of dual information, and hence mainly concerns the continuous relaxations. The Lagrangian relaxation of (P) w.r.t. (2) has an objective function that is still separable in the Hence one can
Computational results
In this section we report results of computational tests of the proposed approach for the Mean–Variance cardinality-constrained portfolio optimization problem on risky assets where is the vector of expected unitary returns, is the prescribed total return, is the variance–covariance matrix, and is the maximum number of purchasable assets. Without the cardinality constraint (), (MV) is well suited for APR: the bound is the same as
Conclusions
The main advantage of the proposed APR technique is its simplicity: just solving (PR) – possibly even approximately with a dual approach – produces the dual solution which can be used to first construct (P+) and then its (APR). Yet, this improves many-fold the performances over plain APR, and even more so over P/C. Notably, APR is quite general and applies to a much larger class than MIQP. It may be worth contrasting 175843 s (P/C in Table 1) with 58 s (APR in Table 3) for
Acknowledgments
The first and third authors acknowledge the contribution of the Italian Ministry for University and Research under the PRIN 2012 Project 2012JXB3YF “Mixed-Integer Nonlinear Optimization: Approaches and Applications”. All the authors acknowledge networking support by the COST Action TD1207. We are grateful to the anonymous referee and the Associate Editor of the Journal for constructive comments that helped us in significantly improve the manuscript.
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