Approximated Perspective Relaxations: a Project&Lift Approach

The Perspective Reformulation (PR) of a Mixed-Integer NonLinear Program with semi-continuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding continuous relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR (P^2R) can be defined where the integer variables are eliminated by projecting the solution set onto the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR (AP^2R) whereby the projected formulation is "lifted" back to the original variable space, with each integer variable expressing one piece of the obtained piecewise-convex function. In some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation, but providing substantially improved bounds. In the process we also substantially extend the approach beyond the original P^2R development by relaxing the requirement that the objective function be quadratic and the left endpoint of the domain of the variables be non-negative. While the AP^2R bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MINLP software; this is shown to be competitive with previously proposed approaches in some applications

Perspective Cuts for the ACOPF with Generators

International audienceThe alternating current optimal power flow problem is a fundamental problem in the management of smart grids. In this paper we consider a variant which includes activation/deactivation of generators at some of the grid sites. We formulate the problem as a mathematical program, prove its NP-hardness w.r.t. ac-tivation/deactivation, and derive two perspective reformulations

Ellipsoidal classification via semidefinite programming

We propose a classification approach exploiting relationships between ellipsoidal separation and Support-vector Machine (SVM) with quadratic kernel. By adding a (Semidefinite Programming) SDP constraint to SVM model we ensure that the chosen hyperplane in feature space represents a non-degenerate ellipsoid in input space. This allows us to exploit SDP techniques within Support-vector Regression (SVR) approaches, yielding better results in case ellipsoid-shaped separators are appropriate for classification tasks. We compare our approach with spherical separation and SVM on some classification problems

A Storm of Feasibility Pumps for Nonconvex MINLP

One of the foremost difficulties in solving Mixed Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programs. Feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems. Such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs: both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed Integer Nonlinear Programming such a property does not hold, and therefore special care has to be exercised in order to allow feasibility pumps algorithms to rely only on local optima of the continuous relaxation. Based on a new, high level view of feasibility pumps algorithms as a special case of the well-known successive projection method, we show that many possible different variants of the approach can be developed, depending on how several different (orthogonal) implementation choices are taken. A remarkable twist of feasibility pumps algorithms is that, unlike most previous successive projection methods from the literature, projection is "naturally" taken in two different norms in the two different subproblems. To cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems; these actually seem to significantly improve the practical performances of the approach. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm

Minimizing power consumption in virtualized cellular networks

Cellular network nodes should be dynamically switched on/off based on the load requirements of the network, to save power and minimize inter-cell interference. This should be done keeping into account global interference effects, which requires a centralized approach. In this paper, we present an architecture, realized within the Flex5GWare EU project, that manages a large-scale cellular network, switching on and off nodes based on load requirements and context data. We describe the architectural framework and the optimization model that is used to decide the activity state of the nodes. We present simulation results showing that the framework adapts to the minimum power level based on the cell loads

Practical feasibility, scalability and effectiveness of coordinated scheduling algorithms in cellular networks towards 5G

Coordinated Scheduling (CS) is used to mitigate inter-cell interference in present (4G) and future (5G) cellular networks. We show that coordination of a cluster of nodes can be formulated as an optimization problem, i.e., placing the Resource Blocks (RB) in each node’s subframe with the least possible over-lapping with neighboring nodes. We provide a clever formulation, which allows optimal solutions to be computed in clusters of ten nodes, and algorithms that compute good suboptimal solutions for clusters of tens of nodes, fast enough for a network to respond to traffic changes in real time. This allows us to assess the relationship between the scale at which CS is performed and its benefits in terms of network energy efficiency and cell-edge user rate. Our results, obtained using realistic power, radiation and Signal-to-Interference-and-Noise-Ratio (SINR) models, show that optimal CS allows a significant protection of cell-edge users. Moreover, this goes hand-in-hand with a reduction in the num-ber of allocated RBs, which in turn allows an operator to reduce its energy consumption. Both benefits actually increase with the size of the clusters. The evaluation is carried out in both a 4G and a foreseen 5G setting, using different power models, system bandwidths and SINR-to-datarate mappings

Solving Electric Market Quadratic Problems by Branch and Fix Coordination Methods

The electric market regulation in Spain (MIBEL) establishes the rules for bilateral and futures contracts in the day-ahead optimal bid problem. Our model allows a price-taker generation company to decide the unit commitment of the thermal units, the economic dispatch of the bilateral and futures contracts between the thermal units and the optimal sale bids for the thermal units observing the MIBEL regulation. The uncertainty of the spot prices is represented through scenario sets. We solve this model on the framework of the Branch and Fix Coordination metodology as a quadratic two-stage stochastic problem. In order to gain computational efficiency, we use scenario clusters and propose to use perspective cuts. Numerical results are reportedPeer Reviewe