Outer Approximation Research Articles

Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems

Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the approximation set up to a multiplicative factor in each component, a convex approximation set only requires this multiplicative approximation to be achieved by some convex combination of finitely many images of solutions in the set. This makes convex approximation sets efficiently computable for a wide range of multiobjective problems: even for many problems for which (classic) approximations sets are hard to compute. In this article, we propose a polynomial-time algorithm to compute convex approximation sets that builds on an exact or approximate algorithm for the weighted sum scalarization and is therefore applicable to a large variety of multiobjective optimization problems. The provided convex approximation quality is arbitrarily close to the approximation quality of the underlying algorithm for the weighted sum scalarization. In essence, our algorithm can be interpreted as an approximate version of the dual variant of Benson’s outer approximation algorithm. Thus, in contrast to existing convex approximation algorithms from the literature, information on solutions obtained during the approximation process is utilized to significantly reduce both the practical running time and the cardinality of the returned solution sets while still guaranteeing the same worst-case approximation quality. We underpin these advantages by the first comparison of all existing convex approximation algorithms on several instances of the triobjective knapsack problem and the triobjective symmetric metric traveling salesman problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was supported by the German Research Foundation [Project 398572517]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0220 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0220 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Tight conic approximation of testing regions for quantum statistical models and measurements

Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of probability distributions; the latter, mapping the space of states to the space of probability distributions. The images of such linear maps are called the testing regions of the corresponding model or measurement. Testing regions are notoriously impractical to treat analytically in the quantum case. Our first result is to provide an implicit outer approximation of the testing region of any given quantum statistical model or measurement in any finite dimension: namely, a region in probability space that contains the desired image, but is defined implicitly, using a formula that depends only on the given model or measurement. The outer approximation that we construct is minimal among all such outer approximations, and close, in the sense that it becomes the maximal inner approximation up to a constant scaling factor. Finally, we apply our approximation to provide sufficient conditions, that can be tested in a semi-device-independent way, for the ability to transform one quantum statistical model or measurement into another.

Numerical conformal bootstrap with analytic functionals and outer approximation

This paper explores the numerical conformal bootstrap in general spacetime dimensions through the lens of a distinct category of analytic functionals, previously employed in two-dimensional studies. We extend the application of these functionals to a more comprehensive backdrop, demonstrating their adaptability and efficacy in general spacetime dimensions above two. The bootstrap is implemented using the outer approximation methodology, with computations conducted in double precision. The crux of our study lies in comparing the performance of this category of analytic functionals with conventional derivatives at crossing symmetric points. It is worth highlighting that in our study, we identified some novel kinks in the scalar channel during the maximization of the gap in two-dimensional conformal field theory. Our numerical analysis indicates that these analytic functionals offer a superior performance, thereby revealing a potential alternative paradigm in the application of conformal bootstrap.

The minimizer of the sum of two strongly convex functions

The optimization problem concerning the determination of the minimizer for the sum of convex functions holds significant importance in the realm of distributed and decentralized optimization. In scenarios where full knowledge of the functions is not available, limiting information to individual minimizers and convexity parameters – either due to privacy concerns or the nature of solution analysis – necessitates an exploration of the region encompassing potential minimizers based solely on these known quantities. The characterization of this region becomes notably intricate when dealing with multivariate strongly convex functions compared to the univariate case. This paper contributes outer and inner approximations for the region harboring the minimizer of the sum of two strongly convex functions, given a constraint on the norm of the gradient at the minimizer of the sum. Notably, we explicitly delineate the boundaries and interiors of both the outer and inner approximations. Intriguingly, the boundaries as well as the interiors turn out to be identical. Furthermore, we establish that the boundary of the region containing potential minimizers aligns with that of the outer and inner approximations.

Risk-averse distributionally robust optimization for construction waste reverse logistics with a joint chance constraint

The uncertainty of the amount of construction and demolition waste (CDW) generation affects the CDW reverse logistics network service level. We investigate a CDW reverse logistics network location-routing problem considering uncertainties. To minimize the total social cost, a two-stage risk-averse distributionally robust optimization model is developed, which aims to optimize the location and number of CDW disposal facilities and the CDW transportation scheme. We introduce the mean-conditional value at risk measure and a joint chance constraint into our model to consider the government’s risk aversion. The above model is approximated as a standard second-order cone programming model (SOCP) considering a special case. To exactly solve the SOCP, we design an outer approximation algorithm. Several performance tests and a case study are proposed, and sensitivity analysis provides helpful managerial insights.

Expanding the reach of quantum optimization with fermionic embeddings

Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.

Nonconvexity and computational effort in the problem of Hydro-Power spillage policy assessment

This paper aims to assess the spillage policies in the Day-Ahead Hydro-Power Plant operation. For this purpose, a combination technique is proposed to prevent erroneous spillage decisions. Given that the developed mathematical modeling lacks the convexity property, we examine the multiple solutions obtained from the Hydro-Power production optimization subject to different power output formulations. In addition, a nonlinear approach to consider friction losses in the modeling is introduced. The modeling is described by Mixed-Integer Linear (MILP) and Integer Nonlinear (MINLP) Programming models. The optimal solutions are provided by Outer Approximation (OA), Branch-and-Bound (BB), and Interior Point algorithms. The simulations show that the water is managed distinctly depending on the optimizer and solution algorithms. Finally, the findings of the spillage policy obtained by the Mixed-Integer Nonlinear model under the cubic function for power generation encourage and indicate its adequacy for a more realistic operation and management of the Mexican Hydro-Power Plant water resources.

Improve operational flexibility of distribution systems using transportable resources

With the continuous progress of carbon neutrality, the accelerating integration of distributed renewable energy sources (RES) poses tremendous pressure on distribution systems' safe and stable operation. The distribution systems that cannot be upgraded in time, e.g., those in border areas, are hard to adapt to the large-scale integration of RES and the growing energy demand, facing a severe lack of operational flexibility. In light of this, the integrated mobile energy station (IMES), which consists of a battery system and photovoltaic cells, is used to improve operational flexibility and alleviate the need for upgrades. An adaptive robust allocation and dispatch model is introduced to coordinate the operation of the distribution system and IMES under multiple uncertainties. The first stage of the proposed model optimizes the locations of IMES using the forecast information of RES and loads, while the second stage optimizes the dispatch plan that is adaptive to the realization of uncertainties. Since the uncertain photovoltaic power in the IMES depends on its deployment location, the proposed model is an adaptive robust optimization with decision-dependent uncertainty, bringing great challenges to the solution. To solve it efficiently, a warmstart-enhanced column-and-constraint generation algorithm is introduced, in which the bi-level subproblem is solved by the dual method and outer approximation algorithm. Finally, case studies demonstrate the effectiveness of the proposed method. The participation of six IMESs can save 5.23 % on the system's total cost and avoid the load shedding caused by transmission congestion, indicating IMES's potential for economical and flexible operation of the system.

An approximation algorithm for multiobjective mixed-integer convex optimization

In this article we introduce an algorithm that approximates the nondominated sets of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches for problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which are multiobjective convex problems, by fixing the integer assignments. The patch problems are solved using (simplicial) Sandwiching. We identify parts of patches that are dominated by other patches and ensure that these patch parts are not refined further. We prove that the algorithm converges and show a bound on the reduction of the approximation error in the course of the algorithm. We illustrate the behaviour of our algorithm using some numerical examples and compare its performance to an algorithm from literature.

Robust dynamic train regulation integrated with stop-skipping strategy in urban rail networks: An outer approximation based solution method

In dense urban rail networks with high passenger demands, uncertain disturbances occur frequently, and the resulting train delays will likely spread over the whole network rapidly, hence degrading the service quality offered to passengers. To cope with the uncertainties of frequent disturbances in urban rail networks, this paper proposes a robust train regulation strategy based on the information gap decision theory, which allows the operators to adjust the conservativeness of adjustment schemes flexibly by varying system performances but without the need for prior knowledge of uncertain disturbances. Specifically, considering the coupling relationship between train dynamic flows and passenger dynamic flows, a mixed integer quadratically constrained programming (MIQCP) model is constructed for the robust train regulation problem to generate solutions with immunity against disturbance uncertainties, in which the envelope-bound model is used to characterizing the uncertain sets of disturbances. To meet the real-time requirements of train operation adjustment, a tailored outer approximation algorithm incorporating a two-phase heuristics method is devised to effectively solve the developed robust train regulation model, thereby quickly generating high-quality solutions. Moreover, the warm start technique and domain reduction technique are carefully developed to accelerate the solving procedure. Numerical experiments based on the Beijing metro network illustrate the robustness of the proposed train regulation strategies and the effectiveness of the designed solution approach.

Dispatchable region for distributed renewable energy generation in reconfigurable AC–DC distribution networks with soft open points

The dispatchable region (DR) describes the ability of a power system to adapt to the variability of distributed renewable energy (DRE) generation. Switchable devices such as soft open points (SOPs) and tie switches enable the AC–DC distribution network to flexibly cope with DRE power fluctuations. Currently, research on DRs seldom considers the effects of SOPs and reconfiguration with tie switches in AC–DC distribution networks, which prevents accurate evaluation of DR for this type of network. To bridge this gap, this paper formulates DRs for DRE generation in an AC–DC distribution network considering SOPs and reconfiguration with tie switches. The original model for deriving DRs is typically nonconvex, and the DR boundaries are intractable because of nonlinear power flow equations and binary variables of tie switches. Accordingly, this paper employs the outer polyhedral approximation and convex hull relaxation to approximate the original DR. An efficient adaptive constraint generation algorithm is developed to search for boundaries of the approximated DR. The numerical results show that the proposed model can effectively adapt to the DR of reconfigurable AC–DC distribution networks with SOPs and verify the accuracy and computational efficiency of the proposed method.

The Terminator: An Integration of Inner and Outer Approximations for Solving Wasserstein Distributionally Robust Chance Constrained Programs via Variable Fixing

We present a novel approach aimed at enhancing the efficacy of solving both regular and distributionally robust chance constrained programs using an empirical reference distribution. In general, these programs can be reformulated as mixed-integer programs (MIPs) by introducing binary variables for each scenario, indicating whether a scenario should be satisfied. Whereas existing methods have focused predominantly on either inner or outer approximations, this paper bridges this gap by studying a scheme that effectively combines these approximations via variable fixing. By checking the restricted outer approximations and comparing them with the inner approximations, we derive optimality cuts that can notably reduce the number of binary variables by effectively setting them to either one or zero. We conduct a theoretical analysis of variable fixing techniques, deriving an asymptotic closed-form expression. This expression quantifies the proportion of binary variables that should be optimally fixed to zero. Our empirical results showcase the advantages of our approach in terms of both computational efficiency and solution quality. Notably, we solve all the tested instances from literature to optimality, signifying the robustness and effectiveness of our proposed approach. History: Accepted by Andrea Lodi/Design & Analysis of Algorithms — Discrete. Funding: This work was supported by Office of Naval Research [N00014-24-1-2066]; Division of Civil, Mechanical and Manufacturing Innovation [2246414]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0299 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0299 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Output-Space Outer Approximation Branch-and-Bound Algorithm for a Class of Linear Multiplicative Programs

Output-Space Outer Approximation Branch-and-Bound Algorithm for a Class of Linear Multiplicative Programs

Deep learning the efficient frontier of convex vector optimization problems

In this paper, we design a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater’s condition. The proposed machine learning methodology provides both an inner and outer approximation of the weakly efficient frontier, as well as an upper bound to the error at each approximated efficient point. In numerical case studies we demonstrate that the proposed algorithm is effectively able to approximate the true weakly efficient frontier of CVOPs. This remains true even for large problems (i.e., many objectives, variables, and constraints) and thus overcoming the curse of dimensionality.

Flexural–torsional buckling with enforced axis of twist and related problems

The classical lateral–torsional instability of a cantilever beam with continuous elastic lateral restraint and a transverse point-load applied at the free end is discussed here through the lens of asymptotic simplifications. One of our main goals is to provide analytical approximations for the critical buckling load, as well as its dependence on various key non-dimensional groups. The first part of this study is concerned with a scenario in which a doubly symmetric beam is constrained to rotate about a fixed axis situated at an arbitrary height above the shear centroidal axis. The second part examines a beam that features a continuous elastic lateral restraint spanning its entire length. Assuming that the stiffness of the constraint, κ (say), is finite, the buckling equations in the second case are described by a system of two coupled fourth-order differential equations in the lateral displacement and the angle of twist. We show that as κ → ∞ , the problem discussed in the first part provides an outer solution for the aforementioned system; the relevant boundary conditions for the next-to-leading-order outer approximation are also derived using matched asymptotics. Our theoretical findings are confirmed by comparisons with direct numerical simulations of the full buckling problem.

Impacts of capacity redundancy and process flexibility on risk mitigation in e-waste recycling supply chain management

Recently Waste Electrical and Electronic equipment (WEEE) has become urgent environmental issues around the world. Generally, the e-waste recycling reverse supply chain (RSC) features higher uncertainties and complicated network structures, making the operation more difficult. Since the study on impact of hybrid uncertainty mitigation strategy in RSC is rare, we study the coupling effect of two strategies — capacity redundancy and process flexibility through the design of two hybrid strategies. A distributionally robust optimization model is proposed to handle both waste returns and resource yields uncertainties, and solved by a proposed Outer Approximation (OA) algorithm after an adaptive robust formulation. With numerical experiments based on real data implemented, our research finds conflict situation of using capacity redundancy and process flexibility strategies together. We also give guidelines for the use of hybrid strategies in RSC. We find that the optimal strategy changes as required service level increases. We also suggest moderate investment in capacity redundancy with long chain flexibility and resource centralization when applying hybrid strategy with optimal flexibility.

Drone-Delivery Network for Opioid Overdose: Nonlinear Integer Queueing-Optimization Models and Methods

This study proposes an emergency network design model that uses a fleet of drones to deliver naloxone in response to opioid overdoses. The network is represented as a collection of M/G/K queueing systems with variable capacity and service time modelled as a decision-dependent random variable. The model is a complex queuing-based optimization problem which locates drone bases and dispatches drones to opioid incidents. The authors devise an efficient solution framework in which they linearize the multiple nonlinearities (fractional, polynomial, exponential, factorial terms), derive an equivalent mixed-integer linear reformulation, and design an outer approximation branch-and-cut algorithm. The authors demonstrate the generalizablity of the approach to any problem minimizing the response time of M/G/K queueing systems with unknown capacity. Tests on Virginia Beach data reveal that drones can decrease response time by 82%, increase survival chance by more than 273%, save up to 33 additional lives annually, and provide up to 279 additional quality-adjusted life years.

Coordinated risk-averse distributionally robust optimization for maintenance and generation schedules of offshore wind farm cluster

The coordinated optimization of maintenance and generation schedules of multiple offshore wind farms (OWFs) bring great benefits in reducing the maintenance and operation cost of OWF cluster. Firstly, a two-stage coordinated risk-averse distributionally robust optimization (CRDRO) model for maintenance and generation schedules of wind turbines (WTs) in OWF cluster is proposed. This model takes into account uncertainties related to the prediction errors in WTs’ maximum available active power outputs, the travel time of maintenance vessels, and the high operational cost risks associated with these uncertainties. A feasible route generating algorithm is designed to obtain all feasible maintenance combinations for WTs and the corresponding routes of maintenance vessels for the CRDRO model. The risk-averse term, which is a weighted sum of expected operation costs and GlueVaR value, is used to measure the tail risk of the worst probability distribution. To solve the two-stage CRDRO model, the risk-averse term is simplified into the form of expected value calculation, which can be further transformed into the tractable form by using the strong duality theorem. Finally, the two-stage CRDRO model is solved by the column-and-constraint generation algorithm, with the nonconvex sub-problem being addressed by the outer approximation algorithm. Case studies on an real-world cluster comprising three OWFs verify the correctness and efficiency of the proposed method, and demonstrate that the CRDRO model enhances the overall economic efficiency and reduces the high operational cost risk under extreme scenarios.

Robust-to-Dynamics Optimization

A robust-to-dynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function [Formula: see text] and a feasible set [Formula: see text]) and (ii) a dynamical system (a map [Formula: see text]). Its goal is to minimize f over the set [Formula: see text] of initial conditions that forever remain in [Formula: see text] under g. The focus of this paper is on the case where the mathematical program is a linear program and where the dynamical system is either a known linear map or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to [Formula: see text]. Our inner approximations are optimized with respect to the objective function f, and their semidefinite characterization—which has a semidefinite constraint of fixed size—is obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations to [Formula: see text] from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case. Funding: O. Günlük was partially supported by the Office of Naval Research [Grant N00014-21-1-2575]. This work was partially funded by the Alfred P. Sloan Foundation, the Air Force Office of Scientific Research, Defense Advanced Research Projects Agency [Young Faculty Award], the National Science Foundation [Faculty Early Career Development Program Award], and Google [Faculty Award].

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm