Outer Approximation Research Articles

Data-adaptive robust coordinated optimization of dynamic active and reactive power flow in active distribution networks

The uncertainty of wind power brings security and economic issues in active distribution networks. As an effective solution to these problems, the active/reactive optimization aims at minimizing the total operation cost and network loss by controlling multiple resources. This paper proposes a two-stage data-adaptive robust optimization model for the dynamic active-reactive power flow. The dynamic correlation between the uncertainties of active and reactive power is analyzed at two kinds of power factor modes of wind turbines and then is described by a linear affine decision rule. To characterize these uncertainties, a data-adaptive multi-band uncertainty set is constructed based on historical data, which allows the number of bands and the corresponding weight coefficients to be adjusted under different confidence levels. Subsequently, the conic relaxation and linearization techniques are performed to reformulate the robust optimization model as a mixed-integer second-order cone programming problem. A two-level solution framework, consisting of a column-and-constraint generation algorithm and an outer approximation algorithm, is employed to solve this problem. Simulation results on a modified IEEE 33-bus system indicate that the variable power factor mode is more suitable for describing the dynamic correlation than the constant power factor mode. Besides, the proposed model and method also show better practicality and solution efficiency than the existing ones.

The supporting hyperplane optimization toolkit for convex MINLP

In this paper, an open-source solver for mixed-integer nonlinear programming (MINLP) problems is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) combines a dual strategy based on polyhedral outer approximations (POA) with primal heuristics. The POA is achieved by expressing the nonlinear feasible set of the MINLP problem with linearizations obtained with the extended supporting hyperplane (ESH) and extended cutting plane (ECP) algorithms. The dual strategy can be tightly integrated with the mixed-integer programming (MIP) subsolver in a so-called single-tree manner, i.e., only a single MIP optimization problem is solved, where the polyhedral linearizations are added as lazy constraints through callbacks in the MIP solver. This enables the MIP solver to reuse the branching tree in each iteration, in contrast to most other POA-based methods. SHOT is available as a COIN-OR open-source project, and it utilizes a flexible task-based structure making it easy to extend and modify. It is currently available in GAMS, and can be utilized in AMPL, Pyomo and JuMP as well through its ASL interface. The main functionality and solution strategies implemented in SHOT are described in this paper, and their impact on the performance are illustrated through numerical benchmarks on 406 convex MINLP problems from the MINLPLib problem library. Many of the features introduced in SHOT can be utilized in other POA-based solvers as well. To show the overall effectiveness of SHOT, it is also compared to other state-of-the-art solvers on the same benchmark set.

Energy Efficiency and Throughput Maximization Using Millimeter Waves–Microwaves HetNets

The deployment of millimeter waves can fulfil the stringent requirements of high bandwidth and high energy efficiency in fifth generation (5G) networks. Still, millimeter waves communication is challenging because it requires line of sight (LOS). The heterogeneous network (HetNet) of millimeter waves and microwaves solves this problem. This paper proposes a millimeter -microwaves heterogeneous HetNet deployed in an indoor factory (InF). In InF, the manufacturing and production are performed inside big and small halls. We consider non standalone dual-mode base stations (DMBS) working on millimeter waves and microwaves. We analyze the network in terms of throughput and energy efficiency (EE). We formulate mixed-integer-non-linear-programming (MINLP) to maximize the throughput and EE of the network. The formulated problem is a complex optimization problem and hard to solve with exhaustive search. We propose a novel outer approximation algorithm (OAA) to solve this problem, and the proposed algorithm OAA achieves optimal solution at β = 10−3. At this β, the average throughput value obtained is approximately 50 Mbps, whereas the value of EE is 4.4 Mbits/J. We also compare the performance of OAA with the mesh-adaptive-direct-search-algorithm (NOMAD), and the experimental results verify that OAA outperforms NOMAD in terms of throughput and EE maximization. We also compare the performance of OAA with particle swarm optimization (PSO), genetic algorithm (GA), and many others optimization algorithms. Experimental results verify that OAA outperforms all other algorithms.

A New Stochastic Model for Bus Rapid Transit Scheduling with Uncertainty

Nowadays, authorities of large cities in the world implement bus rapid transit (BRT) services to alleviate traffic problems caused by the significant development of urban areas. Therefore, a controller is required to control and dispatche buses in such BRT systems.. However, controllers are facing new challenges due to the inherent uncertainties of passenger parameters such as arrival times, demands, alighting fraction as well as running time of vehicles between stops. Such uncertainties may significantly increase the operational cost and the inefficiencies of BRT services. In this paper, we focus on the controller’s perspective and propose a stochastic mixed-integer nonlinear programming (MINLP) model for BRT scheduling to find the optimal departure time of buses under uncertainty. The objective function of the model consists of passenger waiting and traveling time and aims to minimize total time related to passengers at any stop. From the modeling perspective, we propose a new method to generate scenarios for the proposed stochastic MINLP model. Furthermore, from the computational point of view, we implement an outer approximation algorithm to solve the proposed stochastic MINLP model and demonstrate the merits of the proposed solution method in the numerical results. This paper accurately reflect the complexity of BRT scheduling problem and is the first study, to the best of our knowledge, that presents and solves a mixed-integer nonlinear programming model for BRT scheduling.

Computationally Efficient Approximations for Distributionally Robust Optimization Under Moment and Wasserstein Ambiguity

Distributionally robust optimization (DRO) is a modeling framework in decision making under uncertainty in which the probability distribution of a random parameter is unknown although its partial information (e.g., statistical properties) is available. In this framework, the unknown probability distribution is assumed to lie in an ambiguity set consisting of all distributions that are compatible with the available partial information. Although DRO bridges the gap between stochastic programming and robust optimization, one of its limitations is that its models for large-scale problems can be significantly difficult to solve, especially when the uncertainty is of high dimension. In this paper, we propose computationally efficient inner and outer approximations for DRO problems under a piecewise linear objective function and with a moment-based ambiguity set and a combined ambiguity set including Wasserstein distance and moment information. In these approximations, we split a random vector into smaller pieces, leading to smaller matrix constraints. In addition, we use principal component analysis to shrink uncertainty space dimensionality. We quantify the quality of the developed approximations by deriving theoretical bounds on their optimality gap. We display the practical applicability of the proposed approximations in a production–transportation problem and a multiproduct newsvendor problem. The results demonstrate that these approximations dramatically reduce the computational time while maintaining high solution quality. The approximations also help construct an interval that is tight for most cases and includes the (unknown) optimal value for a large-scale DRO problem, which usually cannot be solved to optimality (or even feasibility in most cases). Summary of Contribution: This paper studies an important type of optimization problem, that is, distributionally robust optimization problems, by developing computationally efficient inner and outer approximations via operations research tools. Specifically, we consider several variants of such problems that are practically important and that admit tractable yet large-scale reformulation. We accordingly utilize random vector partition and principal component analysis to derive efficient approximations with smaller sizes, which, more importantly, provide a theoretical performance guarantee with respect to low optimality gaps. We verify the significant efficiency (i.e., reducing computational time while maintaining high solution quality) of our proposed approximations in solving both production–transportation and multiproduct newsvendor problems via extensive computing experiments.

Projectively Self-Concordant Barriers

Self-concordance is the most important property required for barriers in convex programming. It is intrinsically linked to the affine structure of the underlying space. Here we introduce an alternative notion of self-concordance that is linked to the projective structure. A function on a set [Formula: see text] in an affine space is projectively self-concordant if and only if it can be extended to an affinely self-concordant logarithmically homogeneous function on the conic extension of X. The feasible sets in conic programs, notably linear and semidefinite programs, are naturally equipped with projectively self-concordant barriers. However, the interior-point methods used to solve these programs use only affine self-concordance. We show that estimates used in the analysis of interior-point methods are tighter for projective self-concordance, in particular inner and outer approximations of the set. This opens the way to a better tuning of parameters in interior-points algorithms to allow larger steps and hence faster convergence. Projective self-concordance is also a useful tool in the theoretical analysis of logarithmically homogeneous barriers on cones.

Non-convex nested Benders decomposition

We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an varepsilon -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.

Strong convexity of reachable sets of linear systems

The reachable set on some time interval of a linear control system $x'\in Ax {+} U$, $x(0)=0$, is considered. A number of cases is examined when the reachable set is the intersection of some balls of fixed radius $R$ (that is, a strongly convex set of radius $R$). In some cases the radius $R$ is estimated from above. It turns out that strong convexity is fairly typical for this class of reachable sets in a certain sense. Among possible applications of this result are the possibility of constructing outer polyhedral approximation of reachable sets with better accuracy in the Hausdorff metric than in the general case, and applications to linear differential games and some optimization problems. Bibliography: 23 titles.

Facial Input Decompositions for Robust Peak Estimation under Polyhedral Uncertainty

This work bounds extreme values of state functions for a class of input-affine continuous-time systems that are affected by polyhedral-bounded uncertainty. Instances of these systems may arise in data-driven peak estimation, in which the state function must be bounded for all systems that are consistent with a set of state-derivative data records corrupted under L-infinity bounded noise. Existing occupation measure-based methods form a convergent sequence of outer approximations to the true peak value, given an initial set, by solving a hierarchy of semidefinite programs in increasing size. These techniques scale combinatorially in the number of state variables and uncertain parameters. We present tractable algorithms for peak estimation that scale linearly in the number of faces of the uncertainty-bounding polytope rather than combinatorially in the number of uncertain parameters by leveraging convex duality and a theorem of alternatives (facial decomposition). The sequence of decomposed semidefinite programs will converge to the true peak value under mild assumptions (convergence and smoothness of dynamics).

Converging Approximations of Attractors via Almost Lyapunov Functions and Semidefinite Programming

In this paper we combine two existing approaches for approximating global attractors. One of them approximates the global attractors arbitrarily well by sublevel sets related to solutions of infinite dimensional linear programming problems. A downside there is that these sets are not necessarily positively invariant. On the contrary, the second method provides supersets of the global attractor which are positively invariant. Their method on the other hand has the disadvantage that the underlying optimization problem is not computationally tractable without the use of heuristics – and incorporating them comes at the price of losing guaranteed convergence. In this paper we marry both approaches by combining their techniques and we get converging outer approximations of the global attractor consisting of positively invariant sets based on convex optimization via sum-of-squares techniques. The method is easy to use and illustrated by numerical examples.

Zonotopic Kalman Filter-Based Interval Estimation for Discrete-Time Linear Systems With Unknown Inputs

This letter proposes an unknown input zonotopic Kalman filter-based interval observer for discrete-time linear time-invariant systems. In such contexts, a change of coordinates decoupling the state and the unknown inputs is often used. Here, the dynamics are rewritten into a discrete-time linear time-invariant descriptor system by augmenting the state vector with the unknown inputs. A zonotopic outer approximation of the feasible state set is then obtained with a prediction-correction strategy using the information from the system dynamics, known inputs and outputs. Bounds for both the state and unknown inputs are obtained from this zonotopic set. The efficiency of the proposed interval observer is assessed with numerical simulations.

Performance Tradeoff of MVNOs in OFDMA-Based Virtualized Wireless Networks

In this paper, we analyze the tradeoff between the profits gained by mobile virtual network operators (MVNOs) in an orthogonal frequency division multiple access (OFDMA)-based virtualized wireless networks (VWNs). In this respect, MVNOs rent the network resources from a mobile network operator (MNO) to create virtual resources based on allocated rates and the cost due to allocated transmit powers in two different strategies: resource-based isolation strategy and rate-based isolation strategy. In resource-based isolation strategy, it is assumed that the whole bandwidth in each base station is divided equally between MVNOs whereas in rate-based isolation strategy, the whole bandwidth in each base station is allocated dynamically between MVNOs. We then formulate a multi-objective optimization problem (MOOP) to show the tradeoff between the maximum profit of each MVNO. To solve the formulated MOOP, we adopt the weighted Tchebycheff approach, which can provide the complete Pareto optimal region for nonconvex problems. Thus, by developing such formulation, we will be able to find the best profit gained by MVNOs. We also employ monotonic optimization to obtain the global optimal solutions of the non-convex problems by applying polyblock outer approximation algorithm. Our numerical results show the importance of the aforementioned tradeoffs and confirm that the proposed geometric programming (GP) based algorithm achieves excellent performance as compared to the monotonic method.

Energy Efficient Resource Allocation for H-NOMA Assisted B5G HetNets

The resource allocation solution offered based on Non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) schemes are sub-optimal to address the challenging quality of service (QoS) and higher data rate viz-a-viz energy efficiency (EE) requirements in 5th generation (5G) cellular networks. In this work, we maximize the EE using user equipment (UE) clustering (UE-C) with downlink hybrid NOMA (H-NOMA) assisted beyond 5G (B5G) HetNets. We formulate an optimization problem incorporating UE admission in a cluster, UE association with a base station (BS), and power allocation assisted by H-NOMA, i.e., OMA and NOMA schemes in the macro base station (MBS) only and heterogeneous networks (HetNets) environments. The problem formulated is a type of non-linear concave fractional programming (CFP) problem. The Charnes-Cooper transformation (CCT) is applied to the formulated non-linear CFP problem to convert it into a concave optimization, i.e., mixed-integer non-linear programming (MINLP) problem. A two-phase ϵ-optimal outer approximation algorithm (OAA) is used to solve the MINLP problem. The simulation results show that H-NOMA with HetNets outperforms H-NOMA with MBS only in terms of UE admission, UE association, throughput, and EE.

Reconfigurable-Based Fault-Tolerant Control for Continuous-Time Markov Jump Piecewise-Affine Systems with Allowable Switching Paths

In this paper, the reconfigurable control design problem is studied for a class of continuous-time Markov jump piecewise-affine (PWA) systems with sensor and actuator faults. A reconfiguration block is designed to recover mean-square input-to-state stability (ISS) of the faulty closed-loop system under virtual sensor and virtual actuator. Then, a novel algorithm for computing all the allowable switching paths (ASPs) is developed to overcome the issues of redundant empty paths and mismatch regions, and a S-procedure with the ellipsoidal outer approximation estimation is employed to cope with the affine term. Furthermore, a nonlinear approximation is introduced using the PWA technique to approximate the faulty Markov jump nonlinear systems. Finally, one example is illustrated to demonstrate the effectiveness and advantages of the designed reconfigurable control strategy with an ASPs algorithm.

Optimal lot-sizing of an integrated EPQ model with partial backorders and re-workable products: an outer approximation

In this paper, we develop and optimise the lot-sizing policy in an integrated Economic Production Quantity (EPQ) model with partial backorders and re-workable products considering linear and fixed backordering costs. Time intervals, number of lots, and lot size are the decision variables of our model. The cost function includes the set-up costs, the holding costs, the goodwill loss costs, the fixed backorder costs, the backorder costs, the costs of reworking, the production costs, the disposal costs, and the screening costs. The profit function consists of the sale profits obtained from the sold products. The goal is to minimise the cost function and maximise the profit function under stochastic constraints simultaneously. A Lexicographic method is applied for integrating the conflicting objective functions. The integrated objective function, with stochastic constraints, is a Mix Integer Nonlinear Programming (MINLP) model. Accordingly, an Outer Approximation (OA) algorithm is provided for optimal lot-sizing the integrated EPQ model. Two numerical examples and a real large-scale example demonstrate the decent and acceptable performance of the presented OA with respect to the optimality criteria such as optimum solutions, complementarity, and taken iterations.

Integrating stochastic programming and reliability in the optimal synthesis of chemical processes

Plant availability and operating uncertainties are critical considerations for the design and operation of chemical processes as they directly impact service level and economic performance. This paper proposes a two-stage stochastic programming GDP (Generalized Disjunctive Programming) model with reliability constraints to deal with both the exogenous and endogenous uncertainties in process synthesis, where the reliability model is incorporated into the flowsheet superstructure optimization. The proposed stochastic programming model anticipates the market uncertainties through scenarios for selecting the optimal flowsheet topology, equipment sizes and operating conditions, while considering the impact of selecting parallel units for improving plant availability. An improved logic-based outer approximation algorithm is applied to solve the resulting hybrid GDP model, which effectively avoids numerical difficulties with zero flows and provides high quality design solutions. The applicability of the proposed modeling framework and the efficiency of solution strategy are illustrated with two well-known conceptual design case studies: methanol synthesis process and toluene hydrodealkylation process. The model, which integrates reliability (endogenous uncertainty) and exogenous uncertainty, shows the best economic performance with the increasing operational flexibility and plant availability.

Association and Caching in Relay-Assisted mmWave Networks: A Stochastic Geometry Perspective

Limited backhaul bandwidth and blockage effects are two main factors limiting the practical deployment of millimeter wave (mmWave) networks. To tackle these issues, we study the feasibility of relaying as well as caching in mmWave networks. A user association and relaying (UAR) criterion dependent on both caching status and maximum biased received power is proposed by considering the spatial correlation caused by the coexistence of base stations (BSs) and relay nodes (RNs). Using stochastic geometry tools, we decouple the UAR and caching placement issues by analyzing the relationship between UAR probabilities and caching placement probabilities. We then optimize the formulated caching placement problem based on polyblock outer approximation by exploiting the monotonic property in the general case and utilizing convex optimization in the noise-limited case. Accordingly, we propose a BS and RN selection algorithm where caching status at BSs and maximum biased received power are jointly considered. Experimental results demonstrate a significant enhancement of backhaul offloading using the proposed algorithms, and show that deploying more RNs and increasing cache size in mmWave networks is a more cost-effective alternative than increasing BS density to achieve similar backhaul offloading performance.

The efficiency, equity and effectiveness of location strategies in humanitarian logistics: A robust chance-constrained approach

In the preparedness phase of humanitarian logistics, uncertainties from both the supply and demand sides may dramatically increase morbidity and mortality. We consider a distributionally robust facility location model with chance constraints in which the nodes and edges of the network are vulnerable to random failure. Efficiency, effectiveness and equity metrics, which can be explicitly demonstrated as operational costs, service quality and the coverage rate, are incorporated to quantitatively measure system performance under disaster situations. As chance constraints are intractable, we correspondingly propose conic and linear approximations. The reformulated model is solved within the outer approximation framework, where three acceleration techniques, i.e., the branch-and-cut algorithm, in–out algorithm, and Benders decomposition, are embedded to increase the computational efficacy. Through extensive numerical results and a case study, our proposed model is found to be superior to traditional scenario-based approaches.

Airline Network Planning: Mixed-integer non-convex optimization with demand–supply interactions

Airlines routinely use analytics tools to support flight scheduling, fleet assignment, revenue management, crew scheduling, and many other operational decisions. However, decision support systems are less prevalent to support strategic planning. This paper fills that gap with an original mixed-integer non-convex optimization model, named Airline Network Planning with Supply and Demand interactions (ANPSD). The ANPSD optimizes network planning (including route selection, flight frequencies and fleet composition), while capturing interdependencies between airline supply and passenger demand. We first estimate a demand model as a function of flight frequencies and network configuration, using a two-stage least-squares procedure fitted to historical data, and then formalize the ANPSD by integrating the empirical demand function into an optimization model. The model is formulated as a non-convex mixed-integer program. To solve it, we develop an exact cutting plane algorithm, named 2αECP, which iteratively generates hyperplanes to develop an outer approximation of the non-linear demand functions. Computational results show that the 2αECP algorithm outperforms state-of-the-art benchmarks and generates tight solution quality guarantees. A case study based on the network of a major European carrier shows that the ANPSD provides much stronger solutions than baselines that ignore – fully or partially – demand–supply interactions.

Outer Approximation Method for the Unit Commitment Problem with Wind Curtailment and Pollutant Emission

This paper considers the fast and effective solving method for the unit commitment (UC) problem with wind curtailment and pollutant emission in power systems. Firstly, a suitable mixed-integer quadratic programming (MIQP) model of the corresponding UC problem is presented by some linearization techniques, which is difficult to solve directly. Then, the MIQP model is solved by the outer approximation method (OAM), which decomposes the MIQP into a mixed-integer linear programming (MILP) master problem and a nonlinear programming (NLP) subproblem for alternate iterative solving. Finally, simulation results for six systems with up to 100 thermal units and one wind unit in 24 periods are presented, which show the practicality of MIQP model and the effectiveness of OAM.