Exact Optimization Research Articles

Privacy‐Preserving Optimization Algorithm for Distributed Energy Management Over Time‐Varying Graphs: A State Decomposition Method

ABSTRACTThe rapid advancements of intelligent technologies have brought about the potential vulnerability of confidential gradient information linked to cost functions when solving distributed optimization and its related problems. Within the context of distributed energy management, safeguarding such private information has risen to paramount importance. This article investigates a distributed energy management problem (DEMP) to minimize cost while simultaneously satisfying multiple local constraints and protecting the private gradient information of the cost function. To this end, a new privacy‐preserving distributed optimization algorithm under the framework of gradient tracking over time‐varying graphs is proposed for solving the DEMP. Specifically, the auxiliary variables are designed for each node in the algorithm to update the gradient while the original state variables are responsible for the interaction with original neighbors and auxiliary variables. Consequently, the devised algorithm can protect the confidentiality of private cost gradient information, even in the presence of eavesdroppers within the network. In contrast to the homomorphic encryption, signal masking method, and some other algorithms without utilizing the state decomposition, the designed algorithm does not require extra information or computing resources. Moreover, it is proven that the designed algorithm could theoretically converge to the exact optimum of the DEMP at a rate of under some mild assumptions.

Approximate and Exact Optimization Algorithms for the Beltway and Turnpike Problems with Duplicated, Missing, Partially Labeled, and Uncertain Measurements.

The Turnpike problem aims to reconstruct a set of one-dimensional points from their unordered pairwise distances. Turnpike arises in biological applications such as molecular structure determination, genomic sequencing, tandem mass spectrometry, and molecular error-correcting codes. Under noisy observation of the distances, the Turnpike problem is NP-hard and can take exponential time and space to solve when using traditional algorithms. To address this, we reframe the noisy Turnpike problem through the lens of optimization, seeking to simultaneously find the unknown point set and a permutation that maximizes similarity to the input distances. Our core contribution is a suite of algorithms that robustly solve this new objective. This includes a bilevel optimization framework that can efficiently solve Turnpike instances with up to 100,000 points. We show that this framework can be extended to scenarios with domain-specific constraints that include duplicated, missing, and partially labeled distances. Using these, we also extend our algorithms to work for points distributed on a circle (the Beltway problem). For small-scale applications that require global optimality, we formulate an integer linear program (ILP) that (i) accepts an objective from a generic family of convex functions and (ii) uses an extended formulation to reduce the number of binary variables. On synthetic and real partial digest data, our bilevel algorithms achieved state-of-the-art scalability across challenging scenarios with performance that matches or exceeds competing baselines. On small-scale instances, our ILP efficiently recovered ground-truth assignments and produced reconstructions that match or exceed our alternating algorithms. Our implementations are available at https://github.com/Kingsford-Group/turnpikesolvermm.

Exact optimization of inter-array dynamic cable networks for Floating Offshore Wind Farms

Optimization of inter-array dynamic cables for Floating Offshore Wind Farms (FOWFs) using three integer linear programs and a heuristic is presented. Design optimization of fixed-bottom offshore wind is a challenging research problem but the presence of dynamic components in FOWFs adds new complexity - as the Floating Offshore Wind Turbine (FOWT), the support structure including the station-keeping system, and the floating power cables all experience dynamic movement in reaction to wind, wave and even current forcing. In this study, dynamic modelling for the response of this system is first carried out to assess the risk of potential mechanical interference between movable elements. Subsequently, safety zones constraints are defined in the optimization to ensure minimally safe conditions for operation of the combined FOWT/support-structure/cables system. Likewise, additional constraints including maximum thermal limits, tree topology without branching, and others are incorporated. The programs follow an incremental approach. Model 1 proposes a simple way to avoid mechanical interference, Model 2 adds variables modelling mooring lines anchoring, and Model 3 increases the degrees of freedom through addition of the positioning of the touchdown point where the dynamic and static sections meet at the seabed. The applicability is illustrated through realistic case studies for a reference FOWF in Europe. Results show that: (i) Modern branch-and-cut solvers are able to solve Model 2 getting the global optimum in seconds, and (ii) further cost refining can be obtained after wrapping Model 3 in the heuristic, using Model 2 as the initial design, decreasing the cost of this layout by around 1.5% in few hours through a nonrectilinear topology.

An exact and metaheuristic optimization framework for solving Vehicle Routing Problems with Shipment Consolidation using population-based and Swarm Intelligence

The Vehicle Routing Problem with Shipment Consolidation (VRPSC) is a novel variation of the vehicle routing problem that involves multiple commodities, multiple dimensions, a fleet with different types of vehicles, and the challenge of consolidating shipments during the route. Exact algorithms have been suggested to solve the VRPSC problems. Besides exact algorithms, specific metaheuristic algorithms are employed to deliver solutions of superior quality, albeit not necessarily optimal. The Artificial Immune System (AIS) and Genetic Algorithm (GA) are the metaheuristic optimization techniques applied in this research. Genetic programming is included in evolutionary computing, while AIS is included in swarm intelligence. This research presents a vehicle routing model with product consolidation and different product dimensions. These criteria were selected due to their significant impact on the complexity of mathematical problem-solving in VRPSC. The results from applying these metaheuristic algorithms will be compared with those of exact algorithms to compare and analyse different VRPSC solution approaches.

Combining Precision Boosting with LP Iterative Refinement for Exact Linear Optimization

This article studies a combination of the two state-of-the-art algorithms for the exact solution of linear programs (LPs) over the rational numbers in practice, that is, without any roundoff errors or numerical tolerances. By integrating the method of precision boosting inside an LP iterative refinement loop, the combined algorithm is able to leverage the strengths of both methods: the speed of LP iterative refinement, in particular, in the majority of cases when a double-precision floating-point solver is able to compute approximate solutions with small errors, and the robustness of precision boosting whenever extended levels of precision become necessary. We compare the practical performance of the resulting algorithm with both pure methods on a large set of LPs and mixed-integer programs (MIPs). The results show that the combined algorithm solves more instances than a pure LP iterative refinement approach while being faster than pure precision boosting. When embedded in an exact branch-and-cut framework for MIPs, the combined algorithm is able to reduce the number of failed calls to the exact LP solver to zero while maintaining the speed of the pure LP iterative refinement approach. History: Accepted by Antonio Frangioni, Area Editor for Design and Analysis of Algorithms: Continuous. Funding: The work for this article has been conducted within the Research Campus Modal funded by the German Federal Ministry of Education and Research (BMBF) [Grants 05M14ZAM and 05M20ZBM].

Optimized design of horseshoe-and-spur filterless networks leveraging point-to-multipoint coherent pluggable transceivers

Cutting-edge network architectures and solutions are needed to empower operators to address capacity demands in metro and access networks efficiently. The horseshoe topology, along with similar topologies, is commonly employed in metro-aggregation segments due to its compatibility with the hub-and-spoke traffic pattern present in these networks and the survivability that they can provide. A filterless architecture can enhance cost-effectiveness by replacing active elements with passive components. Moreover, supporting coherent-based point-to-multipoint transceivers—enabled by digital subcarrier multiplexing (DSCM)—can yield additional cost savings. It is noteworthy that telecommunication network topologies often evolve to accommodate more end (leaf) nodes, extending the original horseshoe with spurs or small and short trees. This paper targets these types of networks and proposes combining the utilization of coherent pluggable transceivers leveraging DSCM to guarantee transparent communication between the hub and leaf nodes while adopting different filterless node architectures with selective amplifier deployment. Moreover, it discusses the potential advantages of the architecture using an exact optimization framework tailored to various network sizes and scenarios, which accounts for the amplifiers’ placement and the available types of power splitters/combiners/couplers. The results demonstrate that strategically deploying add/drop and express amplifiers, along with optimizing coupler ratios, can effectively meet design requirements while minimizing the number of optical amplifiers required.

Analyzing error probability in aerial full-duplex relay systems: Exact formulations and optimization techniques

In this study, we investigate the performance of aerial full-duplex relay (AFDR) systems. In particular, we examine two practical scenarios: one where there is no direct link between the transmitter and user, and another where such a link exists. For both scenarios, we develop mathematical models to calculate the closed-form expressions of symbol error probabilities (SEPs) for AFDR systems, using a realistic channel aligned with the fifth generation (5G) and beyond yardsticks. To take care of residual self-interference (RSI) in AFDR, we propose an optimal power allocation strategy. Our numerical findings indicate that the performance in the case with a direct link is considerably higher than that in the case without this link. Additionally, we analyze thoroughly the effects of key parameters such as transmit power, RSI levels, carrier frequencies, and AFDR positions. The effect of RSI is significantly strong, and our proposed optimal power allocation method substantially improves system performance, especially in high transmit power scenarios where error floors may occur. Importantly, the optimal power level is dramatically lower than the conventional value where the optimal power cannot be found. Thus, besides reducing the SEPs, optimal power also helps to prolong the time of operation of AFDR. Monte-Carlo simulations are performed to confirm the accuracy of our derived expressions and demonstrate the efficacy of the proposed approaches in practical scenarios.

Exact optimisation of spatiotemporal monitoring networks by p‐splines with applications in groundwater assessment

AbstractThis paper develops methods to optimise the sampling strategy for monitoring networks which have fixed locations with regular sampling but with only a proportion of these locations to be used on each sampling occasion. This creates the need for a dynamic spatiotemporal sampling design which makes optimal choices of the locations to be sampled on each occasion. This is a commonly occurring scenario in many environmental settings where there is an existing network of monitoring stations and sampling can be expensive. The particular context of optimisation of an existing groundwater monitoring network is discussed in the paper. The standard design criteria of integrated variance (IV) and variance of the integral (VI) are adapted to the spatiotemporal setting. p‐spline models are shown to allow exact computation of IV and VI, in the case of the additive errors, and a very good approximation of IV in the case of multiplicative errors. The speed of these exact computations allows the globally optimal sampling design to be identified efficiently. In the standard case of additive errors, the design criteria are able to exploit information across time. The only information needed is the location of sampling points, not the values sampled. This contrasts with the case of multiplicative errors where the design criteria are also influenced by the observed response data. Simulated and real examples are used to illustrate the results throughout.

Improving Temporal Treemaps by Minimizing Crossings

AbstractTemporal trees are trees that evolve over a discrete set of time steps. Each time step is associated with a node‐weighted rooted tree and consecutive trees change by adding new nodes, removing nodes, splitting nodes, merging nodes, and changing node weights. Recently, two‐dimensional visualizations of temporal trees called temporal treemaps have been proposed, representing the temporal dimension on the x‐axis, and visualizing the tree modifications over time as temporal edges of varying thickness. The tree hierarchy at each time step is depicted as a vertical, one‐dimensional nesting relationships, similarly to standard, non‐temporal treemaps. Naturally, temporal edges can cross in the visualization, decreasing readability. Heuristics were proposed to minimize such crossings in the literature, but a formal characterization and minimization of crossings in temporal treemaps was left open. In this paper, we propose two variants of defining crossings in temporal treemaps that can be combinatorially characterized. For each variant, we propose an exact optimization algorithm based on integer linear programming and heuristics based on graph drawing techniques. In an extensive experimental evaluation, we show that on the one hand the exact algorithms reduce the number of crossings by a factor of 20 on average compared to the previous algorithms. On the other hand, our new heuristics are faster by a factor of more than 100 and still reduce the number of crossings by a factor of almost three.

Generating Euler Diagrams Through Combinatorial Optimization

AbstractCan a given set system be drawn as an Euler diagram? We present the first method that correctly decides this question for arbitrary set systems if the Euler diagram is required to represent each set with a single connected region. If the answer is yes, our method constructs an Euler diagram. If the answer is no, our method yields an Euler diagram for a simplified version of the set system, where a minimum number of set elements have been removed. Further, we integrate known wellformedness criteria for Euler diagrams as additional optimization objectives into our method. Our focus lies on the computation of a planar graph that is embedded in the plane to serve as the dual graph of the Euler diagram. Since even a basic version of this problem is known to be NP‐hard, we choose an approach based on integer linear programming (ILP), which allows us to compute optimal solutions with existing mathematical solvers. For this, we draw upon previous research on computing planar supports of hypergraphs and adapt existing ILP building blocks for contiguity‐constrained spatial unit allocation and the maximum planar subgraph problem. To generate Euler diagrams for large set systems, for which the proposed simplification through element removal becomes indispensable, we also present an efficient heuristic. We report on experiments with data from MovieDB and Twitter. Over all examples, including 850 non‐trivial instances, our exact optimization method failed only for one set system to find a solution without removing a set element. However, with the removal of only a few set elements, the Euler diagrams can be substantially improved with respect to our wellformedness criteria.

Exact evaluation and optimization of integer-ratio policies for stochastic one-warehouse multiple-retailer inventory systems

This paper considers one-warehouse multiple-retailer inventory systems. The retailers are non-identical and face Poisson demand processes. We consider an integer-ratio policy in which each retailer orders in a fixed interval that is either an integer or integer-ratio multiple of the warehouse's reorder interval. Base-stock policy and virtual allocation are assumed. We show that cost evaluations at the retailers are completely separable, leading to a simple approach to develop an optimal integer-ratio policy for a two-level distribution system with heterogeneous retailers.

Turkish Cashier Problem with time windows and its solution by matheuristic algorithms

Turkish Cashier Problem (TCP) is a new application area of the traveling salesman problem that was introduced to the literature recently. In this problem, the cashier can use public transportation or take a taxi where the cashier must visit multiple customer locations while minimizing the total transportation cost. In this study, we introduce a more realistic version of this problem where time has been integrated. This aspect is achieved by imposing time intervals within which the cashier must visit the customers. We name this problem as the TCP with time windows (TCPwTW). We develop several matheuristic algorithms to solve the TCPwTW: a modified version of the Simplify and Conquer (SAC) algorithm that was suggested for the TCP, simulated annealing (SA), original and modified versions of the migrating birds optimization (MBO) algorithm coupled with mathematical programming. We also tried to find the exact optimum using a Solver where for complex problems, only lower bounds were found. Numerical experimentation reveals that while for problems with loose time intervals, an exact solver can be considered. Once the time intervals tighten up, the best solutions can be obtained using matheuristics involving SA and MBO.

A comparative study of fuzzy multi-objective investment project portfolio selection and optimization based on financial return and different risk measurements

Competitiveness in the global market is getting more intense. Due to resource and budget constraints, firms need to achieve their expected goals and satisfy all investment constraints under uncertainty. Selecting the set of projects among other candidates to get the most efficient portfolio requires a lot of attention from the Decision Makers (DMs) as this consideration no longer relies purely on the financial term. This problem becomes a multi-objective problem under uncertainty where the financial return and risk from uncertainty are required into the trading off consideration. Due to the financial uncertainty, the chance-constrained programming has been employed in this study for defuzzifying and solving uncertain optimization problems at a specified confidence level that is defined by the DMs. Then, various kinds of investment or financial risk measures, Lower-Semi Variance Index (LSVI), the absolute deviation with the expected FNPV, and the absolute mean-Conditional Value at Risk (CVaR) gap are provided in the selection of such risk measures to show their differences in characteristics and performances in the obtained results. Since, such problems can consist of many project candidates and complex constraints, which may grow beyond the application of the exact optimization approach, a meta-heuristic method, Genetic Algorithm (GA), is introduced to optimize this problem through designing and constructing a decision support tool for the investment portfolio selection and optimization. The applicability of the proposed comparative approach and the constructed tool are illustrated through examples.

Regionalization of primary health care units: An iterated greedy algorithm for large-scale instances

In this paper, we study the problem of multi-institutional regionalization of primary health care units. The problem consists of deciding where to place new facilities, capacity expansions for existing facilities, and demand allocation in a multi-institutional system to minimize the total travel distance from demand points to health care units. It is known that traditional exact methods as branch-and-bound are limited to solving small- to medium-size instances of the problem. Given that real world-instances can be large, in this paper we propose an iterated greedy algorithm with variable neighborhood descent search for handling large-scale instances. Within this solution framework, several methods are developed. A greedy constructive method and two deconstruction strategies are developed. Another interesting component is the exact optimization of a demand allocation subproblem that is obtained when the location of facilities is previously fixed. An empirical assessment using real-world data from the State of Mexico’s Public Health Care System is carried out. The results demonstrate the effectiveness of the proposed metaheuristic in handling large-scale instances.

IMPROVING THE METHODOLOGY FOR OPTIMIZING MULTIMODAL TRANSPORTATION DELIVERY ROUTES AND CYCLIC SCHEDULES IN A TRANSNATIONAL DIRECTION

This study considers the task of planning the routes of multimodal transnational cargo transportation. Due to the extremely long length of such routes, delivery times and costs per cargo unit are extremely important. Delays in various types of transport and in the case of cargo transshipment are associated not only with the growth of cargo flows but also with the inconsistency of vehicle schedules. The purpose of this study is to improve the previously developed methodology for optimizing multimodal cargo transportation, taking into account the need for its application to transnational transport corridors. The content of the formulated network problem is reduced to a modification of the traveling salesman problem with an unknown number of transport points the route should pass through. Such a problem is NP-hard due to the time complexity of the algorithms. A modified algorithm has been developed, according to which the general problem with the number of N points is divided into several sub-problems. Transport points are grouped into consecutive subsets that are related by only one non-alternative way of transportation. This way can be any “bottleneck” of the transport network or an artificially created one. Such a decomposition of the problem gives a set of partial solutions, which were combined into the final optimal solution. The obtained solution to the routing problem of multimodal routes takes into account the cyclical schedules of the transport operation and gives a guaranteed exact optimum for calculations performed within the permissible time. In addition to determining the optimal route, the algorithm makes it possible to determine the required number of vehicles and their work schedules depending on the total cargo flow on the route.

Discovering Sequential Patterns with Predictable Inter-event Delays

Summarizing sequential data with serial episodes allows non-trivial insight into the data generating process. Existing methods penalize gaps in pattern occurrences equally, regardless of where in the pattern these occur. This results in a strong bias against patterns with long inter-event delays, and in addition that regularity in terms of delays is not rewarded or discovered---even though both aspects provide key insight. In this paper we tackle both these problems by explicitly modeling inter-event delay distributions. That is, we are not only interested in discovering the patterns, but also in describing how many times steps typically occur between their individual events. We formalize the problem in terms of the Minimum Description Length principle, by which we say the best set of patterns is the one that compresses the data best. The resulting optimization problem does not lend itself to exact optimization, and hence we propose Hopper to heuristically mine high quality patterns. Extensive experiments show that Hopper efficiently recovers the ground truth, discovers meaningful patterns from real-world data, and outperforms existing methods in discovering long-delay patterns.

Linear convergence of decentralized estimation for statistical estimation using gradient method

There has been a growing interest in solving consensus optimization problems in a multi-agent system. It is known that there is an exactness-speed dilemma for decentralized optimization for the naive gradient method, where either we have fast linear convergence to a neighborhood of size O(η) using a fixed constant step size η, or slow convergence to the exact optimizer with a decreasing step size. We reinvestigate a special case of this problem from a statistical point of view, where each agent’s local function is a sample estimate of a common population function. It is shown that when considering the convergence to the population target instead of to the exact consensual solution, the convergence is linear up to the statistical accuracy of the problem, using a step size only depending on the network characteristics instead of the sample size N. The objective function is not required to be strongly convex (but their statistical limit is), which allows for example the absolute deviation function which is used in median regression. Such results are in stark contrast to the optimization literature, where strong convexity of the objective function is necessary for linear convergence.

Infrared: a declarative tree decomposition-powered framework for bioinformatics.

Many bioinformatics problems can be approached as optimization or controlled sampling tasks, and solved exactly and efficiently using Dynamic Programming (DP). However, such exact methods are typically tailored towards specific settings, complex to develop, and hard to implement and adapt to problem variations. We introduce the Infrared framework to overcome such hindrances for a large class of problems. Its underlying paradigm is tailored toward problems that can be declaratively formalized as sparse feature networks, a generalization of constraint networks. Classic Boolean constraints specify a search space, consisting of putative solutions whose evaluation is performed through a combination of features. Problems are then solved using generic cluster tree elimination algorithms over a tree decomposition of the feature network. Their overall complexities are linear on the number of variables, and only exponential in the treewidth of the feature network. For sparse feature networks, associated with low to moderate treewidths, these algorithms allow to find optimal solutions, or generate controlled samples, with practical empirical efficiency. Implementing these methods, the Infrared software allows Python programmers to rapidly develop exact optimization and sampling applications based on a tree decomposition-based efficient processing. Instead of directly coding specialized algorithms, problems are declaratively modeled as sets of variables over finite domains, whose dependencies are captured by constraints and functions. Such models are then automatically solved by generic DP algorithms. To illustrate the applicability of Infrared in bioinformatics and guide new users, we model and discuss variants of bioinformatics applications. We provide reimplementations and extensions of methods for RNA design, RNA sequence-structure alignment, parsimony-driven inference of ancestral traits in phylogenetic trees/networks, and design of coding sequences. Moreover, we demonstrate multidimensional Boltzmann sampling. These applications of the framework-together with our novel results-underline the practical relevance of Infrared. Remarkably, the achieved complexities are typically equivalent to the ones of specialized algorithms and implementations. Infrared is available at https://amibio.gitlabpages.inria.fr/Infrared with extensive documentation, including various usage examples and API reference; it can be installed using Conda or from source.

Bayesian Optimization via Exact Penalty

Constrained optimization problems pose challenges when the objective function and constraints are nonconvex and their evaluation requires expensive black-box simulations. Recently, hybrid optimization methods that integrate statistical surrogate modeling with numerical optimization algorithms have shown great promise, as they inherit the properties of global convergence from statistical surrogate modeling and fast local convergence from numerical optimization algorithms. However, the computational efficiency is not satisfied by practical needs under limited budgets and in the presence of equality constraints. In this article, we propose a novel hybrid optimization method, called exact penalty Bayesian optimization (EPBO), which employs Bayesian optimization within the exact penalty framework. We model the composite penalty function by a weighted sum of Gaussian processes, where the qualitative components of the constraint violations are smoothed by their predictive means. The proposed method features (i) closed-form acquisition functions, (ii) robustness to initial designs, (iii) the capability to start from infeasible points, and (iv) effective handling of equality constraints. We demonstrate the superiority of EPBO to state-of-the-art competitors using a suite of benchmark synthetic test problems and two real-world engineering design problems.

SkyPIE: A Fast & Accurate Oracle for Object Placement

Cloud object stores offer vastly different price points for object storage as a function of workload and geography. Poor object placement can thus lead to significant cost overheads. Prior cost-saving techniques attempt to optimize placement policies on the fly, deciding object placements for each object individually. In practice, these techniques do not scale to the size of the modern cloud. In this work, we leverage the static nature and pay-per-use pricing model of cloud environments to explore a different approach. Rather than computing object placements on the fly, we precompute a SkyPIE oracle---a lookup structure representing all possible placement policies and the workloads for which they are optimal. Internally, SkyPIE represents placement policies as a matrix of cost-hyperplanes, which we effectively precompute through pruning and convex optimization. By leveraging a fast geometric algorithm, online queries then are 1 to 8 orders of magnitude faster but as accurate as Integer-Linear-Programming. This makes exact optimization tractable for real workloads and we show >10x cost savings compared to state-of-the-art heuristic approaches.