Cone Programming Research Articles

A convex cone programming based implicit material point method

For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To circumvent these limitations, this paper innovatively proposes a convex cone programming-based implicit MPM (CP-MPM) algorithm, which ensures excellent convergence of solving complex problems involving large deformation, regardless of the chosen time step. In the proposed CP-MPM, the governing equations are initially transformed into a stationary point of a multivariable functional, leveraging the generalized Hellinger-Reissner (HR) variational principle. This stationary point problem is subsequently reformulated as a min-max convex cone optimization problem, with constraints originating from elastoplastic constitutive equations. In addition, a novel particle-based adhesive-frictional contact algorithm is proposed to effectively tackle the interaction between MPM domain and rigid bodies. The contact inequality between material points and rigid bodies is transformed into convex cone constraints, which rigorously prevent material point penetration and facilitate the imposition of irregular boundary conditions. Both elastic and elastoplastic problems involving contact under static or dynamic loading are ultimately represented as standard second-order cone programming (SOCP) problems, which is effectively solved by employing the Primal-Dual Interior Point (PDIP) method. The robustness, accuracy and convergence of the proposed method are validated through a series of elastic and elastoplastic benchmark problems. All results demonstrate the CP-MPM is a very promising method for implicitly solving complex practices.

Wasserstein Robust Classification with Fairness Constraints

Problem definition: Data analytics models and machine learning algorithms are increasingly deployed to support consequential decision-making processes, from deciding which applicants will receive job offers and loans to university enrollments and medical interventions. However, recent studies show these models may unintentionally amplify human bias and yield significant unfavorable decisions to specific groups. Methodology/results: We propose a distributionally robust classification model with a fairness constraint that encourages the classifier to be fair in the equality of opportunity criterion. We use a type-[Formula: see text] Wasserstein ambiguity set centered at the empirical distribution to represent distributional uncertainty and derive a conservative reformulation for the worst-case equal opportunity unfairness measure. We show that the model is equivalent to a mixed binary conic optimization problem, which standard off-the-shelf solvers can solve. We propose a convex, hinge-loss-based model for large problem instances whose reformulation does not incur binary variables to improve scalability. Moreover, we also consider the distributionally robust learning problem with a generic ground transportation cost to hedge against the label and sensitive attribute uncertainties. We numerically examine the performance of our proposed models on five real-world data sets related to individual analysis. Compared with the state-of-the-art methods, our proposed approaches significantly improve fairness with negligible loss of predictive accuracy in the testing data set. Managerial implications: Our paper raises awareness that bias may arise when predictive models are used in service and operations. It generally comes from human bias, for example, imbalanced data collection or low sample sizes, and is further amplified by algorithms. Incorporating fairness constraints and the distributionally robust optimization (DRO) scheme is a powerful way to alleviate algorithmic biases. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869] and the Chinese University of Hong Kong [Grant 4055191]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.0230 .

Pilot and data power optimization problems in multiple Aerial Relay Stations cell‐free communication systems

SummaryA cell‐free (CF) technology and unmanned aerial vehicles (UAVs) acting as aerial relay stations (ARSs) are gradually viewed as viable technologies for usage in sixth‐generation (6G) wireless networks owing to their outstanding advantages, such as large coverage, uniform service quality, huge connections, flexible deployment in all geographical areas, and high line‐of‐sight (LoS) probability. As a result, the combination of the CF model and UAVs is suitable for densely populated urban areas, shopping malls, festivals, and locations with complex geography. Furthermore, the integration of the CF model and UAV communication can improve system quality and is considered an entirely new research direction as there is no comprehensive investigation of this combined model. In this study, we investigate the quality of the multi‐ARS CF communication system, where ARSs are equipped with multiple antennas and stochastic distribution within a specific region to simultaneously serve numerous ground users. We establish a closed‐form formulation for the uplink and downlink throughput of individual users by using the matched filtering method and conjugate beamforming technology, respectively. Moreover, we propose a novel method to optimize the pilot and data transmission power coefficients for improving channel estimation and increasing throughput per user, thereby ensuring a high quality of service. The power optimization method is executed via the successive convex approximation (SCA) method, second‐order cone program (SOCP), and linear program. We evaluate system performance according to the cumulative distribution function (CDF) of user throughput based on various parameters, such as the number of users, the number of ARSs, and the length of pilot sequences. The analytical findings also reveal that the system with the proposed power optimization method is always superior to the system without the proposed method in both uplink and downlink.

Interior point algorithm for second-order cone optimization based on a new kernel function

Second-order cone optimization (SOCO) problems are crucial in various fields such as engineering, finance, and machine learning due to their ability to model complex convex optimization tasks efficiently. In this article, we propose an interior point algorithm tailored for SOCO, leveraging a novel kernel function to enhance optimization performance. Traditional interior point methods for SOCO often encounter challenges in scalability and efficiency, particularly with large-scale problems. These methods typically rely on barrier functions that may suffer from slow convergence rates, especially when dealing with highly ill-conditioned or degenerate cones. To address these limitations, our approach introduces a new kernel function that exploits the geometric properties of second-order cones, thereby improving convergence behavior and computational efficiency. The key innovation of our algorithm lies in the construction of the kernel function, which incorporates insights from the structure of second-order cones to efficiently capture the curvature information of the optimization problem. By exploiting this curvature information, our algorithm can effectively navigate the optimization landscape, leading to faster convergence and enhanced robustness compared to traditional methods. Furthermore, our algorithm is equipped with advanced techniques for handling various challenges encountered in SOCO, such as dealing with degenerate cones and exploiting problem structure to accelerate convergence. Additionally, we incorporate strategies for warm-starting and adaptive step-size selection to further improve efficiency, particularly in iterative optimization processes. Finally, our proposed interior point algorithm for second-order cone optimization, based on a new kernel function, derives the iteration bounds O (Nlog (N/∊)) for large update methods. Here denotes N the number of second-order cones in the problem formulation and ∊ the desired accuracy. These iteration bounds are currently the best known bounds for such methods.

The capacitated [formula omitted]-hub interdiction problem with congestion: Models and solution approaches

We study the r-hub interdiction problem under the case of possible congestion. Hub interdiction problems are modeled as attacker-defender problems to identify a set of r critical hubs from a set of p hubs, which when attacked, causes maximum damage to network restoration activities of the defender. In this work we consider that in addition to the routing cost, the defender also aims to minimize the congestion cost. Incorporating the congestion cost in the problem introduces non-linearity in the objective function of the interdiction problem, which makes the problem challenging to solve. To address this, we propose two alternate exact solution approaches. The first approach is an inner-approximation-based approach (IBA), which overestimates the convex non-linear objective function and provides an upper bound. A lower bound is obtained from solving the lower-level problem exactly corresponding to the upper bound solution. The upper bound is tightened using improved approximation with new points generated in successive iterations. In the second approach (referred to as SBA), the problem is reformulated as a second-order conic program, which can be solved using an off-the-shelf solver. From our computational experiments on benchmark datasets (CAB and AP), we demonstrate the efficacy of both the proposed methods. However, IBA consistently outperforms SBA by a significant margin.

Learning Fair Policies for Multi-Stage Selection Problems from Observational Data

We consider the problem of learning fair policies for multi-stage selection problems from observational data. This problem arises in several high-stakes domains such as company hiring, loan approval, or bail decisions where outcomes (e.g., career success, loan repayment, recidivism) are only observed for those selected. We propose a multi-stage framework that can be augmented with various fairness constraints, such as demographic parity or equal opportunity. This problem is a highly intractable infinite chance-constrained program involving the unknown joint distribution of covariates and outcomes. Motivated by the potential impact of selection decisions on people’s lives and livelihoods, we propose to focus on interpretable linear selection rules. Leveraging tools from causal inference and sample average approximation, we obtain an asymptotically consistent solution to this selection problem by solving a mixed binary conic optimization problem, which can be solved using standard off-the-shelf solvers. We conduct extensive computational experiments on a variety of datasets adapted from the UCI repository on which we show that our proposed approaches can achieve an 11.6% improvement in precision and a 38% reduction in the measure of unfairness compared to the existing selection policy.

Hessian barrier algorithms for non-convex conic optimization

AbstractA key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new family of first- and second-order interior-point methods for non-convex optimization problems with linear and conic constraints, combining logarithmically homogeneous barriers with quadratic and cubic regularization respectively. Our approach is based on a potential-reduction mechanism and, under the Lipschitz continuity of the corresponding derivative with respect to the local barrier-induced norm, attains a suitably defined class of approximate first- or second-order KKT points with worst-case iteration complexity $$O(\varepsilon ^{-2})$$ O ( ε - 2 ) (first-order) and $$O(\varepsilon ^{-3/2})$$ O ( ε - 3 / 2 ) (second-order), respectively. Based on these findings, we develop new path-following schemes attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained non-convex optimization problems.

Decisions on ship route, refueling, and sailing speed considering ECA regulation and demand uncertainty

This article aims to maximize the shipping company’s profits by jointly optimizing shipping volume, refueling, ship leg option, and sailing speed with considering of stochastic shipping demand and Emission Control Area (ECA) regulation. Shipping demand is inherently stochastic and intermittent characteristic, causing formidable challenges in shipping service. A distributionally robust optimization method is adopted to handle the demand uncertainties. We build an ambiguity set of shipping demand and introduce a chance constraint with the shipping volume and shipping demand balance to guarantee the satisfactory rate of shipping demand. Moreover, the ECA regulation stimulates the diversification of transportation routes, which further increases transportation uncertainty. Thus, a nonlinear mixed integer programming model with nonconvex chance-constraint is developed. The optimal solution is obtained by transforming the original problem into an attractable second-order conic programming problem. An Asia service route is selected as a numerical experiment. The results show that the proposed model is effective, the optimal refueling strategy, leg option, speed, and shipping volume can be determined simultaneously. Meanwhile, it outperforms the state-of-the-art methods in requiring less demand information and producing fewer conservative results under demand uncertainties. Additionally, some managerial insights are revealed according to the case study.

Robust CARA Optimization

Decision making under uncertainty involves both ambiguity and risk. In “Robust CARA Optimization,” Chen and Sim have developed innovative optimization models designed for ambiguity-averse decision makers whose risk preference is consistent with constant absolute risk aversion (CARA). The research delves into maximizing the worst-case expected exponential utility amid uncertainties from independent factors with ambiguous marginals. To enhance computational feasibility, the authors developed a series of approximations: starting with tractable concave functions in affinely perturbed cases, advancing to concave piecewise affinely perturbed scenarios, and culminating in novel multi-deflected linear decision rules for adaptive optimization. This comprehensive framework extends to a multi-period consumption model, ultimately forming an exponential conic optimization problem efficiently solvable with existing solvers. Practical applications demonstrated in project and multiperiod inventory management highlight the models’ potential to surpass existing stochastic optimization methods, especially under high risk aversion.

Two-stage robust optimization of unified power quality conditioner (UPQC) siting and sizing in active distribution networks considering uncertainty of loads and renewable generators

This paper proposes a robust optimization siting and sizing strategy for the unified power quality conditioner (UPQC) in the active distribution network (ADN) integrated with renewable energy generators. The optimization objectives including investment cost, line loss, and voltage deviation are integrated into an overall cost for both the planning and operating stages. A novel UPQC equivalent power injection model is proposed based on the auxiliary branch method. A set of linear and second-order conic (SOC) constraints are employed to accurately describe the steady-state operation of UPQC. An improved branch current flow model is developed to incorporate radial distribution networks with UPQCs. The UPQC siting and sizing planning is formulated as a mixed integer second-order conic programming (MISOCP) problem. Accounting for the uncertainty in loads and distributed generators, a two-stage robust optimization model is established to address worst-case operational scenarios. A solving strategy is developed based on the strong duality theory and the column-and-constraint generation (C&CG) algorithm, and then commercial solvers can solve the problem accurately and efficiently. Case studies on an IEEE 33-node distribution network demonstrate this strategy's robustness in dealing with loads and renewable energy uncertainties, making it applicable across different UPQC topologies and operation modes. The proposed method has superior performance over the representative meta-heuristic approaches in terms of global optimality as well as an over 7.5 times faster computation speed. This work has significance in enhancing the quality and efficiency of modern power systems highly integrated with renewable energy.

Optimal operational planning of distribution systems: A neighborhood search-based matheuristic approach

This study proposes a strategy for short-term operational planning of active distribution systems to minimize operating costs and greenhouse gas (GHG) emissions. The strategy incorporates network reconfiguration, switchable capacitor bank operation, dispatch of fossil fuel-based and renewable distributed energy resources, energy storage devices, and a demand response program. Uncertain operational conditions, such as energy costs, power demand, and solar irradiation, are addressed using stochastic scenarios derived from historical data through a k-means technique. The mathematical formulation adopts a stochastic scenario-based mixed-integer second-order conic programming (MISOCP) model. To handle the computational complexity of the model, a neighborhood-based matheuristic approach (NMA) is introduced, employing reduced MISOCP models and a memory strategy to guide the optimization process. Results from 69 and 118-node distribution systems demonstrate reduced operational costs and GHG emissions. Moreover, the proposed NMA outperforms two commercial solvers. This work provides insights into optimizing the operation of distribution systems, yielding economic and environmental benefits.

Appointment Scheduling with Delay Tolerance Heterogeneity

In this study, we investigate an appointment sequencing and scheduling problem with heterogeneous user delay tolerances under service time uncertainty. We aim to capture the delay tolerance effect with heterogeneity, in an operationally effective and computationally tractable fashion, for the appointment scheduling problem. To this end, we first propose a Tolerance-Aware Delay (TAD) index that incorporates explicitly the user tolerance information in delay evaluation. We show that the TAD index enjoys decision-theoretical rationale in terms of Tolerance sensitivity, monotonicity, and convexity and positive homogeneity, which enables it to incorporate the frequency and intensity of delays over the tolerance in a coherent manner. Specifically, the convexity of TAD index ensures a tractable modeling of the collective delay dissatisfaction in the appointment scheduling problem. Using the TAD index, we then develop an appointment model with known empirical service time distribution that minimizes the overall tolerance-aware delays of all users. We analyze the impact of delay tolerance on the sequence and schedule decisions and show that the resultant TAD appointment model can be reformulated as a mixed-integer linear program (MILP). Furthermore, we extend the TAD appointment model by considering service time ambiguity. In particular, we encode into the TAD index a moment ambiguity set and a Wasserstein ambiguity set, respectively. The former captures effectively the correlation among service times across positions and user types, whereas the latter captures directly the service time data information. We show that both of the resultant TAD models under ambiguity can be reformulated as polynomial-sized, mixed-integer conic programs (MICPs). Finally, we compare our TAD models with some existing counterpart approaches and the current practice using synthetic data and a case of real hospital data, respectively. Our results demonstrate the effectiveness of the TAD appointment models in capturing the user delay tolerance with heterogeneity and mitigating the worst-case delays. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: S. Wang was supported by the National Natural Science Foundation of China [Grants 71922020, 72171221, and 71988101, entitled “Econometric Modeling and Economic Policy Studies”], the Fundamental Research Funds for the Central Universities [Grant UCAS-E2ET0808X2], and the Major Program of National Natural Science Foundation of China [Grant 72192843]. S. Wang was also supported by a grant from MOE Social Science Laboratory of Digital Economic Forecasts and Policy Simulation at UCAS. 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.2023.0025 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0025 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A penalty-type method for solving inverse optimal value problem in second-order conic programming

This paper aims to consider a type of inverse optimal value problem in second-order conic programming, in which the parameter in its objective function needs to be adjusted under a given class that makes the corresponding optimal objective value closest to a target value. This inverse problem can be reformulated as a minimization problem with some second-order cone complementarity constraints. To tackle these bilinear constraints, we apply a penalty-type method and show that the associated penalty term is exact, which avoids the hurdles of penalty-type methods in the update strategy of the penalty parameter. Numerical results show that our method is suitable to solve the given inverse optimal value problem under different metric distances between the optimal value of the forward problem and a target objective counterpart.

Coordinated planning of DGs and soft open points in multi‐voltage level distributed networks based on the Stackelberg game

AbstractThere are significant differences in distributed generators (DGs) distribution and load characteristics between different voltage levels, which makes it difficult to match sources and loads. We focus on the problem of different consumption capacities at different voltage levels and the divergence of interests among investment entities, and propose a coordinated planning model for DGs and soft open points (SOPs) based on the Stackelberg game. Firstly, a model of a multi‐voltage level distribution networks (DNs) is constructed based on SOPs. Next, the source‐load matching degree is proposed as a measure of the degree of matching between sources and loads in the DN, and the source‐load consumption rate is selected as an indicator to evaluate the impact of the load on DG consumption. Following this, the interest demands of DG investors and distribution company (DisCo) in multi‐voltage levels DN are analyzed, a planning mode based on the Stackelberg game is proposed, and this is solved by combining the genetic algorithm with second‐order cone programming. Finally, the effectiveness of the planning model is tested and verified using an improved IEEE 28‐node system. The results show that the proposed model improves the DG consumption capacity of DNs with multiple voltage levels while protecting the interests of DG investors and DisCo.

The Pine Cone Optimization Algorithm (PCOA).

The present study introduces a novel nature-inspired optimizer called the Pine Cone Optimization algorithm (PCOA) for solving science and engineering problems. PCOA is designed based on the different mechanisms of pine tree reproduction, including pollination and pine cone dispersal by gravity and animals. It employs new and powerful operators to simulate the mentioned mechanisms. The performance of PCOA is analyzed using classic benchmark functions, CEC017 and CEC2019 as mathematical problems and CEC2006 and CEC2011 as engineering design problems. In terms of accuracy, the results show the superiority of PCOA to well-known algorithms (PSO, DE, and WOA) and new algorithms (AVOA, RW_GWO, HHO, and GBO). The results of PCOA are competitive with state-of-the-art algorithms (LSHADE and EBOwithCMAR). In terms of convergence speed and time complexity, the results of PCOA are reasonable. According to the Friedman test, PCOA's rank is 1.68 and 9.42 percent better than EBOwithCMAR (second-best algorithm) and LSHADE (third-best algorithm), respectively. The authors recommend PCOA for science, engineering, and industrial societies for solving complex optimization problems.

Evaluation of the energy performance of a shallow geothermal facility through a second‐order cone programming

AbstractEnergy consumption and operational cost are two decision‐making factors which evaluate the feasibility of any energy system. Whether a thermal unit is efficiently working or not depends on the state properties of the thermodynamic system. Therefore, operational research on a geothermal site was performed. Most research pivots around the basic energy conservation equation to determine the parameters. However, finding the parameters does not serve the purpose of an investor. Investment of capital in any project depends on the return and feasibility of the task performed under the project. In the same direction, this article focuses on the optimisation of the performance of a shallow geothermal site in terms of energy and plant economics using the concept of a second‐order cone program (SOCP). The site was used to operate the hybrid heating, ventilation, and air conditioning unit (HVAC) installed in an academic building. The second‐order cone programming was considered to minimize the energy requirement and cost of energy generation during the winter season. It is to be noted that the solver‐based approach was considered to examine the unit. The optimized energy consumption of GSHP was 24.29 kW⋅h. The levelized cost of energy (LCOE/kW⋅h) derived from SOCP would vary from $ 0.38 to $ 0.58 for the shallow geothermal plant. The rate of return for air conditioning in the building through the geothermal plant would be 1.74%. The optimized upfront cost per m2 was estimated to be $ 1440.87.

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.

Supply Restoration of Data Centers in Flexible Distribution Networks With Spatial-Temporal Regulation

The reliable operation of Internet data centers (IDCs) is crucial to digitization transformation of society. However, the extreme fault on the IDC-located distribution network is fatal to IDC operation. The evolution of flexible distribution networks (FDNs) can satisfy the deep de-pendence of IDC on high-quality and reliable power supply. In FDNs, AC/DC feeders with multiple voltage levels can be flexibly interconnected and adjusted by flexible stations (FSs). It is promising to flexibly enhance operation per-formance of crucial IDCs after fault isolation in FDNs. In this paper, a coordinated supply restoration method of IDCs in FDNs with spatial-temporal regulation is proposed to minimize IDC data-dropping loss under complex faults. First, a spatial-temporal transfer strategy of IDC work-loads is precisely formulated and an FS-based network reconfiguration strategy is proposed. Then, a mixed-integer second-order conic programming model is established for efficient solving. Spatial-temporal coordination among the source, network, storage, and demand sides can be realized without violating physical feasibility. Finally, the proposed method is validated on the modified IEEE 33-node and modified IEEE 123-node test cases. Case studies show that total outage loss is effectively reduced with sufficient utili-zation of available power. Operational performance of FDN is also enhanced under extreme faults.

Attitude Trajectory Planning for Spacecraft With Time-Varying Mass Using Sequential Conic Optimization

Attitude Trajectory Planning for Spacecraft With Time-Varying Mass Using Sequential Conic Optimization

Optimising hierarchical emergency logistics network under ambiguous demand and transportation cost

This article studies the response problem of an emergency logistics network with a decision hierarchy relationship under uncertainty. To account for the partial distribution information about uncertain demand and transportation costs, we construct a moment-based ambiguity set based on limited historical data, where the pivot variable method is employed to determine the confidence interval of the mean value. Based on the constructed ambiguity set, we develop a novel distributionally robust bi-level post-disaster emergency logistics location-routeing model. By exploiting the structural characteristic, chance-constrained models under box-ellipsoid and budget perturbation sets are reformulated as bi-level mixed-integer conic programming models. To accelerate the solution procedure, the bi-level models are further converted into single-level ones via Karush-Kuhn-Tucker condition, which can be directly solved to optimality using CPLEX software. Supply risk value for each supplier is obtained by applying analytic hierarchy process. We conduct extensive experiments using the Iranian flood as the case study to address the computational performance of our proposed optimisation method.