Cone Programming Research Articles

Revealing Vulnerability of N-1 Secure Power Systems to Coordinated Cyber-Physical Attacks

Coordinated cyber-physical (CCP) attacks have attracted wide attention in power systems because of their potential to cause severe disturbance including cascading failures. However, as realistic power systems operate within the N-1 security criterion, existing CCP attacks may be in part mitigated. As such, this paper analyzes impacts of CCP attacks on the vulnerability of the power systems that employ the N-1 security constrained optimal power flow (SCOPF). Specifically, a tri-level model is proposed to analyze the CCP attack impacts, whereby the adversary coordinates a physical attack with cyber attacks to initiate and propagate post-contingency overload based on N-1 SCOPF. A methodology utilizing semidefinite programming (SDP) relaxation and primal-dual formulation is proposed to transform the tri-level model into a conic optimization, such that the model can be easily solved by SDP solvers. Case studies on the IEEE 14, 57, and 118 bus test systems demonstrate that the CCP attacks in power systems with N-1 security criterion are able to cause N-1-1 contingency and even trigger cascading failures.

Optimal Configuration of Coupled Optical Storage Hydrogen Production System Considering Multi-body and Multi-objective Joint Optimization

With the development of siting and capacity setting methods for coupled optical-storage hydrogen production systems, difficulties such as the complexity of constraints in multi-body and multi-objective system planning models, the decline of electrolytic tank life, and non-linear and non-time-series have emerged one after another. To address these difficulties, this paper proposes a double-layer planning capacity configuration method for coupled optical-storage-grid hydrogen production for transmission and transformer systems, which solves the explicit and implicit coupled double-layer optimization problems under the mixed-integer optimization framework with traditional cone optimization techniques combined with swarm intelligent optimization techniques to achieve joint optimal planning configuration for coupled optical-storage-hydrogen production systems.

On second order cone programming approach to two-stage network data envelopment analysis

In two recent papers, [Guo, C., Wei, F., & Chen, Y. (2017). A note on second order cone programming approach to two-stage network data envelopment analysis. European Journal of Operational Research, 263(2), 733–735; and Chen, K., & Zhu, J. (2017). Second order cone programming approach to two-stage network data envelopment analysis. European Journal of Operational Research, 262(1), 231238] converted the general multiplicative two-stage DEA (data envelopment analysis) network model into a parametric/single second order conic programming (SOCP) problem. In this paper, we (i) note that the proposed transformation cannot result in an equivalent mathematical program in view of the fact that the general multiplicative two-stage DEA model is basically a nonconvex optimisation problem, possibly with several local and global optimums, whereas a single SOCP is a convex optimisation; (ii) demonstrate that there is a subtle technical error contained in the main transformation process proposed in both papers; (iii) illustrate, by providing a numerical counterexample, that the optimal solution obtained from the resulting single SOCP problem might be substantially different from the original model’s optimum; and (iv) offer some computational remark for solving the multiplicative/additive two-stage DEA model.

A new second‐order conic optimization model for the Euclidean Steiner tree problem in Rd$\mathbb {R}^d$

AbstractIn this work, a new second‐order conic optimization model for the Euclidean Steiner tree problem in d‐space, where the dimension d is at least three, will be presented. Also, two strategies for eliminating isomorphic trees will be developed. Computational results highlighting the efficiency of this model compared to existing models for the Euclidean Steiner tree problem will be discussed. Finally, enforcing the proposed model with any of the isomorphic tree elimination strategies permits us to compute for the first time, to the best of our knowledge, the minimum Steiner tree for the cube in .

Two‐stage distributionally robust optimization model for a pharmaceutical cold supply chain network design problem

AbstractPharmaceutical safety has received increasing attention from governments and corporations, and building a safe and effective pharmaceutical cold supply chain network has become an important issue. This paper proposes a two‐stage pharmaceutical cold supply chain network design problem considering drug safety, in which the drug demand, transportation costs, and drug safety risk costs are assumed to be random variables. Due to the presence of uncertainty and the fact that information about the distribution of uncertain parameters is often only partially known, a distributionally robust optimization method is used to handle the uncertainty. A two‐stage distributionally robust optimization model is constructed, in which the reliability of the estimation of the demand necessary to satisfy the entire cold chain network is ensured to be greater than a certain predetermined level by introducing an ambiguous joint chance constraint. The decision process for the problem can be divided into strategic and operational decisions with the goal of minimizing the total costs related to facility construction, the purchase of raw materials, drug production, transportation, and safety risks. By introducing an ambiguity set with mean and covariance information to describe the uncertain parameters, the two‐stage model is eventually reformulated as a standard second‐order cone program, thus making it computationally tractable. Finally, numerical experiments are presented to demonstrate the effectiveness of the proposed models and optimization methods.

Branch-and-price algorithms for large-scale mission-oriented maintenance planning problems

This paper presents a novel column-generation-based approach for solving large-scale instances of the joint selective maintenance and repairperson assignment problem (JSM-RAP) for mission-oriented systems in industrial settings. Such systems perform consecutive missions separated by scheduled finite-duration breaks during which some of their components are imperfectly maintained by repairpersons, aiming to maximize system reliability in subsequent missions. The resulting mathematical model is computationally expensive, even for problems of moderate size. The proposed approach decomposes the JSM-RAP into a master problem and multiple subproblems that are solved to generate maintenance patterns, i.e., columns. Two methods are developed to handle the mixed-integer nonlinear subproblems: a piecewise-linear approximation and an exact reformulation into mixed-integer exponential conic programs. Branch-and-price algorithms are developed by embedding the column-generation method into a branch-and-bound tree to restore solution integrality and guarantee its optimality. Furthermore, we use a stabilization scheme to accelerate convergence. Numerical experiments validate the proposed approach and demonstrate its added value in terms of computation time and solution quality. Problem instances of very-large size, similar to real-life industrial production plants, are solved efficiently. Results also show that increasing the number of maintenance levels grants more flexibility to the optimizer to find combinations of components and maintenance actions that better use the limited resources.

A novel robust Volt/Var optimization method based on worst‐case scenarios for distribution network operation

AbstractThe Volt/Var optimization (VVO) problem is used for scheduling the voltage regulation and reactive power compensation equipment in distribution networks to minimize power loss and voltage violation. In order to solve the VVO problem for a forthcoming time horizon, it is necessary to predict some parameters such as load demand and renewable energy production. The prediction of these parameters is always accompanied by uncertainty that robust optimization can be used to solve this concern. This paper presents a scenario‐based robust Volt/Var optimization (RVO) method that significantly reduces the number of scenarios required for the worst‐case approach. Solving the VVO problem with the worst‐case scenarios reduces the computational burden and maintains the voltage security of the distribution network against the severe events. The proposed RVO is formulated based on a mixed‐integer second‐order cone programming (MISOCP) model in which Volt/Var control (VVC) equipment is scheduled over a two‐stage strategy. The proposed method is validated using modified 33‐bus and 69‐bus IEEE test systems. The results demonstrate that the proposed RVO method maintains the network's voltage profile within the acceptable range against uncertainties.

Centralized-local PV voltage control considering opportunity constraint of short-term fluctuation

This study proposes a two-stage photovoltaic (PV) voltage control strategy for centralized control that ignores short-term load fluctuations. In the first stage, a deterministic power flow model optimizes the 15-minute active cycle of the inverter and reactive outputs to reduce network loss and light rejection. In the second stage, the local control stabilizes the fluctuations and tracks the system state of the first stage. The uncertain interval model establishes a chance constraint model for the inverter voltage-reactive power local control. Second-order cone optimization and sensitivity theories were employed to solve the models. The effectiveness of the model was confirmed using a modified IEEE 33 bus example. The intraday control outcome for distributed power generation considering the effects of fluctuation uncertainty, PV penetration rate, and inverter capacity is analysed.

A globally convergent smoothing Newton method for the second order cone complementarity approach of elastoplasticity problems

A new algorithm for solving small-deformation elastoplasticity problems as second order cone complementarity problems (SOCCPs) is presented. The SOCCP is derived from the Hellinger–Reissner variational principle, and then solved with a smoothing Newton method that is modified from the globally convergent Qi-Sun-Zhou method. The method is compared with two well-established methods: the return-mapping method and the conic programming (CP) approach with primal–dual interior-point method. Numerical results indicate that the proposed method shares common characteristics of CP class methods and offers better convergence efficiency in some cases. Moreover, it is shown that the proposed method can be imbedded in the framework of conventional finite element codes with only a few modifications, which makes it easy-to-implement and adaptive to large-scale parallel computing.

Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes

Solving large scale nonlinear optimization problems requires either significant computing resources or the development of specialized algorithms. For Linear Programming (LP) problems, decomposition methods can take advantage of problem structure, gradually constructing the full problem by generating variables or constraints. We first present a direct adaptation of the Column Generation (CG) methodology for nonlinear optimization problems, such that when optimizing over a structured set $${\mathcal {X}}$$ plus a moderate number of complicating constraints, we solve a succession of (1) restricted master problems on a smaller set $${\mathcal {S}}\subset {\mathcal {X}}$$ and (2) pricing problems that are Lagrangean relaxations wrt the complicating constraints. The former provides feasible solutions and feeds dual information to the latter. In turn, the pricing problem identifies a variable of interest that is then taken into account into an updated subset $${\mathcal {S}}'\subset {\mathcal {X}}$$ . Our approach is valid whenever the master problem has zero Lagrangean duality gap wrt to the complicating constraints, and not only when $${\mathcal {S}}$$ is the convex hull of the generated variables as in CG for LPs, but also with a variety of subsets such as the conic hull, the linear span, and a special variable aggregation set. We discuss how the structure of $${\mathcal {S}}$$ and its update mechanism influence the number of iterations required to reach near-optimality and the difficulty of solving the restricted master problems, and present linearized schemes that alleviate the computational burden of solving the pricing problem. We test our methods on synthetic portfolio optimization instances with up to 5 million variables including nonlinear objective functions and second order cone constraints. We show that some CGs with linearized pricing are 2–3 times faster than solving the complete problem directly and are able to provide solutions within 1% of optimality in 6 h for the larger instances, whereas solving the complete problem runs out of memory.

Wasserstein distributionally robust chance-constrained program with moment information

This paper studies a distributionally robust joint chance-constrained program with a hybrid ambiguity set including the Wasserstein metric, and moment and bounded support information of uncertain parameters. For the considered mathematical program, the random variables are located in a given support space, so a set of random constraints with a high threshold probability for all the distributions that are within a specified Wasserstein distance from an empirical distribution, and a series of moment constraints have to be simultaneously satisfied. We first demonstrate how to transform the distributionally robust joint chance-constrained program into an equivalent reformulation, and show that such a program with binary variables can be equivalently reformulated as a mixed 0–1 integer conic program. To reduce the computational complexity, we derive a relaxed approximation of the joint DRCCP-H using McCormick envelop relaxation, and introduce linear relaxed and conservative approximations by using norm-based inequalities when the Wasserstein metric uses the lp-norm with p≠1 and p≠∞. Finally, we apply this new scheme to address the multi-dimensional knapsack and surgery block allocation problems. The results show that the model with a hybrid ambiguity set yields less conservative solutions when encountering uncertainty over the model with an ambiguity set involving only the Wasserstein metric or moment information, verifying the merit of considering the hybrid ambiguity set, and that the linear approximations significantly reduce the computational time while maintaining high solution quality.

A CVaR optimization method for priority of hesitant fuzzy preference relation with chance constraint

In this paper, we establish a chance constrained model for the priority of hesitant fuzzy preference relation based on the idea of statistical distribution for preference information as stochastic variables with unknown distribution. Inspired by the idea of conditional value-at-risk (CVaR) robust optimization, a deterministic convex reformulation is proposed for tackling the chance constrained problem. The existing state-of-the-art methods usually assume that the probability density function of preference information is known a priori, such as Gaussian distribution. However, it is generally over-conservatism. On the contrary, our proposed method provides a tractable second-order cone (SOC) reformulation for the chance constrained problem with the first and second moments, which is easy to handle and calculate. We also analyze the weight acquisition problem of hesitant fuzzy preference relation with unknown distribution preference using the SOC programming method, and obtain the priority weight with its approximately equivalent computationally tractable conic optimization model. A case study is conducted which shows that the proposed method achieves a good general conclusion by comparing it with the optimization method under Gaussian distribution. In addition, this method can also get better decision support for incomplete preference information.

Real-Time Sequential Conic Optimization for Multi-Phase Rocket Landing Guidance

We introduce a multi-phase rocket landing guidance framework that can handle nonlinear dynamics and does not mandate any additional mixed-integer or nonconvex constraints to handle discrete temporal events/switching. To achieve this, we first introduce sequential conic optimization (seco), a new paradigm for solving nonconvex optimal control problems that is entirely devoid of matrix factorizations and inversions. This framework combines sequential convex programming (SCP) and first-order conic optimization and can solve unified multi-phase trajectory optimization problems in real-time. The novel features of this framework are: (1) time-interval dilation, which enables multi-phase trajectory optimization with free-transition-time; (2) single-crossing compound state-triggered constraints, which are entirely convex if the trigger and constraint conditions are convex; (3) virtual state, which is a new approach to handling artificial infeasibility in SCP methods that preserves the shapes of the constraint sets; and, (4) the use of the proportional-integral projected gradient method (pipg), a high-performance first-order conic optimization solver, in tandem with the penalized trust region (ptr) SCP algorithm. We demonstrate the efficacy and real-time capability of seco by solving a relevant multi-phase rocket landing guidance problem with nonlinear dynamics and convex constraints only, and observe that our solver is 2.7 times faster than a state-of-the-art convex optimization solver.

Extrapolated Proportional-Integral Projected Gradient Method for Conic Optimization

Conic optimization is the minimization of a convex quadratic function subject to conic constraints. We introduce a novel first-order method for conic optimization, named <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">extrapolated proportional-integral projected gradient method (xPIPG)</i> , that automatically detects infeasibility. The iterates of xPIPG either asymptotically satisfy a set of primal-dual optimality conditions, or generate a proof of primal or dual infeasibility. We demonstrate the application of xPIPG using benchmark problems in model predictive control. xPIPG outperforms many state-of-the-art conic optimization solvers, especially when solving large-scale problems.

Optimal Scheduling and Policy Design of Two-Dose Vaccination Rollout

In many countries, the COVID-19 vaccination program faces great challenges, from the management of limited and irregular supply of vaccines, uncertain vaccine take-up rate from the population, and appropriate appointment booking management to reduce congestion etc. In addition, the feature of a two-dose regimen of most COVID-19 vaccines poses a unique operational challenge, the "blocking phenomenon," where the need to reserve vaccines for the second-dose appointment may "block" the take-up rate for the first-dose appointment. Determining the appropriate volume of vaccines to be kept in reserve in case of disruption to the supply schedule is an important operational problem for vaccine rollout. In this paper, we use the concept of "booking curve" (from the revenue management literature) to develop a practical tool that jointly determines the vaccine appointment booking limits that control the administration of the first and second doses, followed by an invitation schedule that decision-makers can use to regulate the appointment bookings (demand). The optimization framework aims to design the vaccine rollout policy to maximize the vaccination rate, while ensuring that the aggregate appointment waiting time in the system remains minimal, with sufficient cushion to account for supply disruption and uncertain take-up of appointments. We show that the optimization problems can be efficiently solved by linear and conic programs. A vaccine rollout numerical study based on the Singapore vaccination program is presented to demonstrate the novelty and advantage of the optimization framework. The optimization framework has been developed into an open-source tool to assist the policymakers in designing an effective and adaptive COVID-19 vaccine rollout policy facing the evolving challenges in fighting against the pandemic.

A revised sequential quadratic semidefinite programming method for nonlinear semidefinite optimization

In 2020, Yamakawa and Okuno proposed a stabilized sequential quadratic semidefinite programming (SQSDP) method for solving, in particular, degenerate nonlinear semidefinite optimization problems. The algorithm is shown to converge globally without a constraint qualification, and it has some nice properties, including the feasible subproblems, and their possible inexact computations. In particular, the convergence was established for approximate-Karush-Kuhn-Tucker (AKKT) and trace-AKKT conditions, which are two sequential optimality conditions for the nonlinear conic contexts. However, recently, complementarity-AKKT (CAKKT) conditions were also considered, as an alternative to the previous mentioned ones, that is more practical. Since few methods are shown to converge to CAKKT points, at least in conic optimization, and to complete the study associated to the SQSDP, here we propose a revised version of the method, maintaining the good properties. We modify the previous algorithm, prove the global convergence in the sense of CAKKT, and show some preliminary numerical experiments.

On second-order conic programming duals for robust convex quadratic optimization problems

In this paper, we deal with second-order conic programming (SOCP) duals for a robust convex quadratic optimization problem with uncertain data in the constraints. We first introduce a SOCP dual problem for this robust convex quadratic optimization problem with polytopic uncertain sets. Then, we obtain a zero duality gap result between this robust convex quadratic optimization problem and its dual problem in terms of a new robust type characteristic cone constraint qualification. We also construct a SOCP dual problem for this robust convex quadratic optimization problem with norm-constrained uncertain sets and obtain the corresponding zero duality gap result between them. Moreover, some numerical examples are given to explain the obtained results.

A Unified Early Termination Technique for Primal-Dual Algorithms in Mixed Integer Conic Programming

We propose an early termination technique for mixed integer conic programming within branch-and-bound based solvers. Our approach generalizes previous early termination results for ADMM-based solvers to a broader class of primaldual algorithms, including both operator splitting and interior point methods. The complexity for checking early termination is O(n) for each termination check assuming a bounded problem domain. We show that this domain restriction can be relaxed for problems whose data satisfies a simple rank condition, in which case each check requires an O(n2) solve using a linear system that is factored only once at the root node. We further show how this approach can be used in hybrid model predictive control problems with bounded inputs. Numerical results show that our method leads to a moderate reduction in the computational time required for branch-and-bound conic solvers with interior-point based subsolvers.

Fast Solving Method for Two-Stage Multi-Period Robust Optimization of Active and Reactive Power Coordination in Active Distribution Networks

This paper considers constraints on the cycle number of energy storage systems (ESSs), travel distances of switchable capacitors reactors (SCRs), on load tap changers (OLTCs) and step voltage regulators (SVRs). The characteristics and operational constraints of these equipments are properly formulated with binaries and big-M method, obtaining desirable linear descriptions. The branch flow equations of distribution systems for active and reactive power coordination are constructed. Considering physical constraints, a multi-period mixed-integer deterministic second-order conic programming (SOCP) to minimize network losses is formulated. Then, considering uncertainties of loads and active power of renewable distributed generators (DGs), a two-stage robust model is developed. A directly iteratively solving method between the first and second stage models is proposed. In contrast to the traditional column-and-constraint generation (CCG) method, increases to variables and constraints are not needed to solve the first stage model. To solve the second stage multi-period model, only the model of each single period needs to be solved first. Then their objective function values are accumulated, and the worst scenarios of each period are concatenated. As a result, the solving complexity is greatly reduced. The capabilities of the proposed method are validated by three simulation cases. It is found that the computational rate using the proposed method is significantly enhanced with much less computer storage. Specifically, the simulation results of 4, 33 and 69-bus systems indicate that the computing rate of the proposed method is about 28 times higher than CCG method when 8 iterations are performed. Meanwhile, the precision of optimization results is also improved.

Dehomogenization for completely positive tensors

A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the dehomogenization approach.