Perspective Reformulation Research Articles

Mixed Integer Optimization of the Flow Pattern for Stochastic Feeder Reconfiguration

Mixed-integer conic programming provides an approach to solving the Optimal Feeder Reconfiguration (OFR) problem with guarantees on the quality of the solution. Integrating renewable generation into the distribution network and its associated variability renders stochastic OFR, which simultaneously considers several snapshots of the network, a more viable approach. While the established mixed-integer conic program for deterministic OFR can be extended to the stochastic case, the resulting optimization problem that still uses the bus injection relaxation is challenging to solve. This paper proposes a new mixed integer conic optimization of the flow pattern for solving the stochastic OFR. The new optimization framework exploits the perspective reformulation to obtain a tighter relaxation for stochastic OFR, which improves computational performance, and a feasibility pump heuristic to give a feasible solution. The mixed integer optimization of the flow pattern can also provide an initial solution to the mixed integer conic program employing the bus injection relaxation, giving rise to a hierarchical solution approach. Numerical results on stochastic OFR show that the hierarchical approach provides much-improved system performance compared to solutions considering a single snapshot. In addition, the proposed feasibility pump heuristic gives rise to network configurations close to global optimality in several test instances.

Comparing perspective reformulations for piecewise-convex optimization

We aim at generalizing formulations for non-convex piecewise-linear problems to mathematical programs whose non-convexities are only expressed in terms of piecewise-convex univariate functions. This is motivated by solving Mixed-Integer Non-Linear Programming (MINLP) problems with separable non-convex functions via the Sequential Convex MINLP technique. We theoretically and computationally compare different formulations, showing that, unlike in the linear case, they are not equivalent when perspective reformulation is applied to strengthen the formulation of each single segment.

A new perspective on low-rank optimization

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable relaxations. We invoke the matrix perspective function—the matrix analog of the perspective function—to characterize explicitly the convex hull of epigraphs of simple matrix convex functions under low-rank constraints. Further, we combine the matrix perspective function with orthogonal projection matrices—the matrix analog of binary variables which capture the row-space of a matrix—to develop a matrix perspective reformulation technique that reliably obtains strong relaxations for a variety of low-rank problems, including reduced rank regression, non-negative matrix factorization, and factor analysis. Moreover, we establish that these relaxations can be modeled via semidefinite constraints and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization and leads to new relaxations for a broad class of problems.

An Alternating Method for Cardinality-Constrained Optimization: A Computational Study for the Best Subset Selection and Sparse Portfolio Problems

Cardinality-constrained optimization problems are notoriously hard to solve in both theory and practice. However, as famous examples, such as the sparse portfolio optimization and best subset selection problems, show, this class is extremely important in real-world applications. In this paper, we apply a penalty alternating direction method to these problems. The key idea is to split the problem along its discrete-continuous structure to obtain two subproblems that are much easier to solve than the original problem. In addition, the coupling between these subproblems is achieved via a classic penalty framework. The method can be seen as a primal heuristic for which convergence results are readily available from the literature. In our extensive computational study, we first show that the method is competitive to a commercial mixed-integer program solver for the portfolio optimization problem. On these instances, we also test a variant of our approach that uses a perspective reformulation of the problem. Regarding the best subset selection problem, it turns out that our method significantly outperforms commercial solvers and it is at least competitive to state-of-the-art methods from the literature. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: The first author thanks the Deutsche Forschungsgemeinschaft (DFG) for its support within “Algorithmic Optimization” [Grant RTG 2126]. The third author thanks the DFG for its support within project A05 and B08 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modeling, Simulation and Optimization using the Example of Gas Networks.” Supplemental Material: The supplementary material is available at https://doi.org/10.1287/ijoc.2022.1211 .

Necessary and Sufficient Conditions for Rank-One-Generated Cones

A closed convex conic subset [Formula: see text] of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of [Formula: see text] is closely related to the exactness of semidefinite program (SDP) relaxations of nonconvex quadratically constrained quadratic programs (QCQPs) related to [Formula: see text]. We consider the case where [Formula: see text] is obtained as the intersection of the PSD cone with finitely many homogeneous linear matrix inequalities and conic constraints and identify sufficient conditions that guarantee that [Formula: see text] is ROG. Our general framework allows us to recover a number of well-known results from the literature. In the case of two linear matrix inequalities, we also establish the necessity of our sufficient conditions. This extends one of the few settings from the literature—the case of one linear matrix inequality and the S-lemma—where an explicit characterization for the ROG property exists. Finally, we show how our ROG results on cones can be translated into inhomogeneous SDP exactness results and convex hull descriptions in the original space of a QCQP. We close with a few applications of these results; specifically, we recover the well-known perspective reformulation of a simple mixed-binary set via the ROG toolkit. Funding: This work was supported by the Office of Naval Research [Grant N00014-19-1-2321] with the National Science Foundation (NSF) Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI-1454548] and Frank A. and Helen E. Risch Faculty Development Chair. Part of this work was done while the second author was visiting the Simons Institute for the Theory of Computing, which was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through the NSF Division of Computing and Communication Foundations [Grant CCF-1740425].

Ideal formulations for constrained convex optimization problems with indicator variables

Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultaneously consider the nonlinear non-separable objective, indicator variables, and combinatorial constraints. Specifically, we give the convex hull description of the epigraph of the composition of a one-dimensional convex function and an affine function under arbitrary combinatorial constraints. As special cases of this result, we derive ideal convexifications for problems with hierarchy, multi-collinearity, and sparsity constraints. Moreover, we also give a short proof that for a separable objective function, the perspective reformulation is ideal independent from the constraints of the problem. Our computational experiments with sparse regression problems demonstrate the potential of the proposed approach in improving the relaxation quality without significant computational overhead.

Gaining or losing perspective

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $$x\in \{0\}\cup [l,u]$$ , where z is a binary indicator of $$x\in [l,u]$$ ( $$u> \ell > 0$$ ), and y “captures” f(x), which is assumed to be convex on its domain [l, u], but otherwise $$y=0$$ when $$x=0$$ . This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the “perspective reformulation” inequality $$y \ge zf(x/z)$$ . We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $$f(x) := x^p$$ , for $$p>1$$ , relaxations utilizing the inequality $$yz^q \ge x^p$$ , for $$q \in [0,p-1]$$ , are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $$f(x) := x^2$$ ) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.

Decompositions of Semidefinite Matrices and the Perspective Reformulation of Nonseparable Quadratic Programs

We study the problem of decomposing the Hessian matrix of a mixed integer convex quadratic program (MICQP) into the sum of positive semidefinite 2 × 2 matrices. Solving this problem enables the use of perspective reformulation techniques for obtaining strong lower bounds for MICQPs with semicontinuous variables but a nonseparable objective function. An explicit formula is derived for constructing 2 × 2 decompositions when the underlying matrix is weakly scaled diagonally dominant, and necessary and sufficient conditions are given for the decomposition to be unique. For matrices lying outside this class, two exact semidefinite programming approaches and an efficient heuristic are developed for finding approximate decompositions. We present preliminary results on the bound strength of a 2 × 2 perspective reformulation for the portfolio optimization problem, showing that, for some classes of instances, the use of 2 × 2 matrices can significantly improve the quality of the bound with respect to the best previously known approach, although at a possibly high computational cost.

Green hub location problem

This paper introduces the green hub location problem that finds the best locations for hubs, assignments of demand nodes to these hubs and speed of trucks/flights so as to route the demand between any origin-destination pairs. The aim of the service provider is to minimize the total amount of emissions that depends on vehicle speed and payload while routing the deliveries within a predetermined service time limit. In this study, we first propose a nonlinear model for this problem, which is then reformulated as a second order cone programming formulation. We strengthen the new model by using perspective reformulation approach. An extensive computational study on the CAB and TR datasets demonstrates the benefits of incorporating green transportation service activities to the classic hub location problems. We also provide insights for the carrier companies by analyzing the solutions with different discount factors, service time limits and number of hubs.

Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In order to overcome this shortcoming, we reformulate the problem using the perspective of the quadratic function. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve convex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed algorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they provide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LP-based branch-and-bound algorithm with extended formulations. The software that was reviewed as part of this submission was given the Digital Object identifier https://doi.org/10.5281/zenodo.1489153 .

Strengthening the sequential convex MINLP technique by perspective reformulations

The sequential convex MINLP (SC-MINLP) technique is a solution method for nonconvex mixed-integer nonlinear problems (MINLPs) where the nonconvexities are separable. It is based on solving a sequence of convex MINLPs which trade a better and better relaxation of the nonconvex part of the problem with the introduction of more and more piecewise-linear nonconvex terms, and therefore binary variables. The convex MINLPs are obtained by partitioning the domain of each separable nonconvex term in the intervals in which it is convex and those in which it is concave. In the former, the term is left in its original form, while in the latter it is piecewise-linearized. Since each interval corresponds to a semi-continuous variable, we propose to modify the convex terms using the Perspective Reformulation technique to strengthen the bounds. We show by means of experimental results on different classes of instances that doing so significantly decreases the solution time of the convex MINLPs, which is the most time consuming part of the approach, and has therefore the potential to improving the overall effectiveness of SC-MINLP.

Strong formulations for quadratic optimization with M-matrices and indicator variables

We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each, and provide the convex-hull descriptions of these sets. We also describe strong conic quadratic valid inequalities. Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.

Review on Reformulation of the Mean-Variance Model with Real-life Trading Restrictions

In this paper, we consider a class of portfolio selection problems with cardinality and minimum buy-in threshold constraints in real-life which can be formulated as mixed-integer quadratic programming (MIQP). Two reformulation methods that generate the same tight continuous relaxation of original problem are compared in the context under the branch-and-bound algorithm, one is the Perspective Reformulation and another is the Lift-and-Convexification Reformulation (LCR). Computational results show that the (PC) is more competitive than the (LCR) method in terms of computing time and nodes in MIQP solver CPLEX 12.7, what's more, this outperformance becomes more obvious as the size of instances grows.

Improving the Approximated Projected Perspective Reformulation by dual information

We propose an improvement of the Approximated Projected Perspective Reformulation (AP2R) for dealing with constraints linking the binary variables. The new approach solves the Perspective Reformulation (PR) once, and then use the corresponding dual information to reformulate the problem prior to applying AP2R, thereby combining the root bound quality of the PR with the reduced relaxation computing time of AP2R. Computational results for the cardinality-constrained Mean–Variance portfolio optimization problem show that the new approach is competitive with state-of-the-art ones.

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

We consider in this paper a class of semi-continuous quadratic programming problems which arises in many real-world applications such as production planning, portfolio selection and subset selection in regression. We propose a lift-and-convexification approach to derive an equivalent reformulation of the original problem. This lift-and-convexification approach lifts the quadratic term involving $x$ only in the original objective function $f(x,y)$ to a quadratic function of both $x$ and $y$ and convexifies this equivalent objective function. While the continuous relaxation of our new reformulation attains the same tight bound as achieved by the continuous relaxation of the well known perspective reformulation, the new reformulation also retains the linearly constrained quadratic programming structure of the original mix-integer problem. This prominent feature improves the performance of branch-and-bound algorithms by providing the same tightness at the root node as the state-of-the-art perspective reformulation and offering much faster processing time at children nodes. We further combine the lift-and-convexification approach and the quadratic convex reformulation approach in the literature to form an even tighter reformulation. Promising results from our computational tests in both portfolio selection and subset selection problems numerically verify the benefits from these theoretical features of our new reformulations.

Optimal Energy-Efficient Transmit Beamforming for Multi-User MISO Downlink

This paper studies beamforming techniques for energy efficiency maximization (EEmax) in multiuser multiple-input single-output (MISO) downlink system. For this challenging nonconvex problem, we first derive an optimal solution using branch-and-reduce-and-bound (BRB) approach. We also propose two low-complexity approximate designs. The first one uses the well-known zero-forcing beamforming (ZFBF) to eliminate inter-user interference so that the EEmax problem reduces to a concave-convex fractional program. Particularly, the problem is then efficiently solved by closed-form expressions in combination with the Dinkelbach's approach. In the second design, we aim at finding a stationary point using the sequential convex approximation (SCA) method. By proper transformations, we arrive at a fast converging iterative algorithm where a convex program is solved in each iteration. We further show that the problem in each iteration can also be approximated as a second-order cone program (SOCP), allowing for exploiting computationally efficient state-of-the-art SOCP solvers. Numerical experiments demonstrate that the second design converges quickly and achieves a near-optimal performance. To further increase the energy efficiency, we also consider the joint beamforming and antenna selection (JBAS) problem for which two designs are proposed. In the first approach, we capitalize on the perspective reformulation in combination with continuous relaxation to solve the JBAS problem. In the second one, sparsity-inducing regularization is introduced to approximate the JBAS problem, which is then solved by the SCA method. Numerical results show that joint beamforming and antenna selection offers significant energy efficiency improvement for large numbers of transmit antennas.

Approximated perspective relaxations: a project and 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 ($$\mathrm{P}^2\mathrm{R}$$P2R) 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 ($$\mathrm{AP}^2\mathrm{R}$$AP2R) 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 $$\mathrm{P}^2\mathrm{R}$$P2R 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 $$\mathrm{AP}^2\mathrm{R}$$AP2R 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.

Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints that arise naturally in various real-world applications such as portfolio selection and subset selection in regression. This class of problems can be formulated as mixed-integer 0-1 quadratic programs. We propose a new semidefinite program (SDP) approach for computing the “best” diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation of the problem. We also give an alternative way of deriving the perspective reformulation by applying a special Lagrangian decomposition scheme to the diagonal decomposition of the problem. This derivation can be viewed as a “dual” method to the convexification method employing the perspective function on semicontinuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of mixed-integer quadratic programming solvers when applied to the perspective reformulations of the problem.

Delay-Constrained Shortest Paths: Approximation Algorithms and Second-Order Cone Models

Routing real-time traffic with maximum packet delay in contemporary telecommunication networks requires not only choosing a path but also reserving transmission capacity along its arcs, as the delay is a nonlinear function of both components. The problem is known to be solvable in polynomial time under quite restrictive assumptions, i.e., equal rate allocations (all arcs are reserved the same capacity) and identical reservation costs, whereas the general problem is $$\mathcal {NP}$$ NP -hard. We first extend the approaches to the equal rate allocation (ERA) version to a pseudo-polynomial Dynamic Programming one for integer arc costs and a FPTAS for the case of general arc costs. We then show that the general problem can be formulated as a mixed-integer Second-Order Cone (SOCP) program and therefore solved with off-the-shelf technology. We compare two formulations: one based on standard big-M constraints and one where Perspective Reformulation techniques are used to tighten the continuous relaxation. Extensive computational experiments on both real-world networks and randomly generated realistic ones show that the ERA approach is fast and provides an effective heuristic for the general problem whenever it manages to find a solution at all, but it fails for a significant fraction of the instances that the SOCP models can solve. We therefore propose a three-pronged approach that combines the fast running time of the ERA algorithm and the effectiveness of the SOCP models, and show that it is capable of solving realistic-sized instances with high accuracy at different levels of network load in a time compatible with real-time usage in an operating environment.

Tight polyhedral approximation for mixed-integer linear programming unit commitment formulations

The perspective reformulation has been recently proposed for constructing tight relaxations of the unit commitment (UC) problem with quadratic cost functions. In this case, it has been shown that the perspective reformulation can be cast as a second-order cone program. Conic quadratic programming is based on interior-point methods and does not benefit from the warm-start capabilities of the simplex method available in commercial mixed-integer linear programming software. It is known that the perspective formulation can be approximated with arbitrary accuracy by dynamically adding gradient-type inequalities. This study presents a higher dimensional polyhedral approximation of the perspective reformulation. The proposed approximation is not based on gradient information and can be easily implemented in a modelling language; in addition, it requires defining additional variables and constraints whose number grows moderately with increasing levels of accuracy. Extensive numerical comparisons show that the polyhedral approximation of the perspective reformulation yields a tight UC formulation, which is competitive with currently known ones.