Chance-constrained Problem Research Articles

On Optimal Orchestration of Virtualized Cellular Networks With Downlink Rate Coverage Probability Constraints

Wireless network virtualization is emerging as a promising technology for next-generation (5G) cellular networks. A key advantage of introducing virtualization in cellular networks is that wireless services can be decoupled from network resources (e.g., infrastructure and spectrum) so that multiple virtual networks can be built using a shared pool of network resources. This paper develops a theoretical framework for optimizing the resource allocation in virtualized cellular networks with heterogeneous coverage requirements. Specifically, we first formulate a chance-constrained virtual resource allocation problem that aims at probabilistically guaranteeing virtual networks’ downlink coverage and rate demand satisfaction while minimizing resource over-provisioning in the presence of uncertainty in user equipment (UE) locations and channel conditions. Thereafter, we derive a closed-form expression for the downlink rate coverage probability of a typical virtual network. With the closed-form expression, we design an efficient algorithm to solve the chance-constrained problem with affordable computation complexity. Furthermore, considering the possibility of lack of sufficient network resources to satisfy all virtual networks’ demands, we design a prioritized virtual resource allocation scheme where virtual networks are built sequentially based on their given priorities. Our results demonstrate that the proposed stochastic virtualization framework outperforms existing deterministic virtualization frameworks in terms of probabilistically guaranteeing virtual networks’ coverage and rate demand satisfaction.

Solving Chance-Constrained Problems via a Smooth Sample-Based Nonlinear Approximation

We introduce a new method for solving nonlinear continuous optimization problems with chance constraints. Our method is based on a reformulation of the probabilistic constraint as a quantile functi...

Nonlinear chance-constrained problems with applications to hydro scheduling

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

A scalable stochastic programming approach for the design of flexible systems

We study the problem of designing systems in order to minimize cost while meeting a given flexibility target. Flexibility is attained by enforcing a joint chance constraint, which ensures that the system will exhibit feasible operation with a given target probability level. Unfortunately, joint chance constraints are complicated mathematical objects that often need to be reformulated using mixed-integer programming (MIP) techniques. In this work, we cast the design problem as a conflict resolution problem that seeks to minimize cost while maximizing flexibility. We propose a purely continuous relaxation of this problem that provides a significantly more scalable approach relative to MIP methods and show that the formulation delivers solutions that closely approximate the Pareto set of the original joint chance-constrained problem.

Investment optimization of grid-scale energy storage for supporting different wind power utilization levels

With the large-scale integration of renewable generation, energy storage system (ESS) is increasingly regarded as a promising technology to provide sufficient flexibility for the safe and stable operation of power systems under uncertainty. This paper focuses on grid-scale ESS planning problems in transmission-constrained power systems considering uncertainties of wind power and load. A scenario-based chance-constrained ESS planning approach is proposed to address the joint planning of multiple technologies of ESS. Specifically, the chance constraints on wind curtailment are designed to ensure a certain level of wind power utilization for each wind farm in planning decision-making. Then, an easy-to-implement variant of Benders decomposition (BD) algorithm is developed to solve the resulting mixed integer nonlinear programming problem. Our case studies on an IEEE test system indicate that the proposed approach can co-optimize multiple types of ESSs and provide flexible planning schemes to achieve the economic utilization of wind power. In addition, the proposed BD algorithm can improve the computational efficiency in solving this kind of chance-constrained problems.

Optimal black start strategy for microgrids considering the uncertainty using a data‐driven chance constrained approach

Microgrids may suffer from full blackouts when confronted with unexpected disruptions due to man-made faults or natural disasters. How to quickly restore the power supply of microgrids by making use of local distributed energy resources (DERs) is therefore a practical issue to help microgrids ride through full blackouts. Accordingly, this study proposes a novel black start strategy for microgrids to determine the restoration sequence and optimally allocate the DERs after full blackouts. In particular, the uncertainty of power output of renewable energy sources is considered using a data-driven chance-constrained approach when renewable energy sources are integrated. The proposed approach only utilises historical data and does not need any prior knowledge about the true probability distribution of the uncertainty. In addition, affine-based techniques and sample-dependent band functions are developed to convert the chance-constrained problem into a tractable mixed-integer linear programming problem. Finally, numerical experiments based on two microgrid test systems are performed to validate the effectiveness of the proposed model.

A robust chance constraint programming approach for evacuation planning under uncertain demand distribution

This study focuses on an evacuation planning problem where the number of actual evacuees (demand) is unknown at the planning phase. In the context of mass evacuation, we assume that only partial information about the demand distribution (i.e., moment, support, or symmetry) is known as opposed to the exact distribution in a stochastic environment. To address this issue, robust approximations of chance-constrained problems are explored to model traffic demand uncertainty in evacuation networks. Specifically, a distributionally robust chance-constrained model is proposed to ensure a reliable evacuation plan (start time, path selection, and flow assignment) in which the vehicle demand constraints are satisfied for any probability distribution consistent with the known properties of the underlying unknown evacuation demand. Using a path-based model, the minimum clearance time is found for the evacuation problem under partial information of the random demand. Numerical experiments show that the proposed approach works well in terms of solution feasibility and robustness as compared to the solution provided by a chance constrained programming model under the assumption that the demand distribution follows a known probability distribution.

Combined Stochastic Optimization of Frequency Control and Self-Consumption With a Battery

Optimally combining frequency control with self-consumption can increase revenues from battery storage systems installed behind-the-meter. This work presents an optimized control strategy that allows a battery to be used simultaneously for self-consumption and primary frequency control. Therein, it addresses two stochastic problems: the delivery of primary frequency control with a battery and the use of the battery for self-consumption. We propose a linear recharging policy to regulate the state of charge of the battery while providing primary frequency control. Formulating this as a chance-constrained problem, we can ensure that the risk of battery constraint violations stays below a predefined probability. We use robust optimization as a safe approximation to the chance-constraints, which allows to make the risk of constraint violation arbitrarily low, while keeping the problem tractable and offering maximum reserve capacity. Simulations with real frequency measurements prove the effectiveness of the designed recharging strategy. We adopt a rule-based policy for self-consumption, which is optimized using stochastic programming. This policy allows to reserve more energy and power of the battery on moments when expected consumption or production is higher, while using other moments for recharging from primary frequency control. We show that optimally combining the two services increases value from batteries significantly.

An approach to chance constrained problems using weighted empirical distribution and differential evolution with application to flood control planning

AbstractThis paper proposes a new approach to solve chance constrained problems (CCPs) efficiently. Specifically, the probabilistic constraint in CCP can be evaluated directly if the cumulative distribution function (CDF) of uncertain function value is known. Therefore, the CDF is approximated by using weighted empirical CDF (W ECDF). Then a powerful evolutionary algorithm, namely differential evolution, combined with W ECDF is used to solve CCP. In order to demonstrate the performance of the proposed method, it is applied to flood control planning problems.

An algorithm for binary linear chance-constrained problems using IIS

We propose an algorithm based on infeasible irreducible subsystems to solve binary linear chance-constrained problems with random technology matrix. By leveraging on the problem structure we are able to generate good quality upper bounds to the optimal value early in the algorithm, and the discrete domain is used to guide us efficiently in the search of solutions. We apply our methodology to individual and joint binary linear chance-constrained problems, demonstrating the ability of our approach to solve those problems. Extensive numerical experiments show that, in some cases, the number of nodes explored by our algorithm is drastically reduced when compared to a commercial solver.

Distributionally Robust Secure Transmission for MISO Downlink Networks With Assisting Jammer

In this paper, we propose the distributionally robust secure transmit schemes in multi-input single-output (MISO) downlink wireless networks which consist of a transmitter, a desired receiver, multiple eavesdroppers and an assisting jammer. The imperfect channel state information (CSI) is considered and the CSI errors are only captured by the mean and covariance. We first study the transmit power minimization by jointly designing the beamforming vector at transmitter and artificial noise(AN) covariance at jammer, while the lower bound of connection probability at desired receiver and lower bound of outage probability at eavesdroppers are guaranteed simultaneously. Since the chance-constrained problem is non-convex, we derive two safe convex approximations by exploiting the Conditional Value-at-risk (CVaR) based method and Bernstein-type inequality (BTiE) based method, respectively. Specifically, we extend the application of BTiE originally proposed with the Gaussianity assumption to other possible distributions which satisfy the proposed sufficient condition. Furthermore, the secrecy rate maximization is exploited under the constraint of total transmit power. The original problem is non-convex and fractional, hence we design the Bilevel Quick Search (BQS) method to make it tractable. Finally, the simulation results verify the effectiveness and the robustness of the proposed transmit schemes.

An Approach to Chance Constrained Problems using Weighted Empirical Distribution and Differential Evolution With Application to Flood Control Planning

An Approach to Chance Constrained Problems using Weighted Empirical Distribution and Differential Evolution With Application to Flood Control Planning

Risk-sensitive semi-Markov decision processes with general utilities and multiple criteria

Abstract In this paper we investigate risk-sensitive semi-Markov decision processes with a Borel state space, unbounded cost rates, and general utility functions. The performance criteria are several expected utilities of the total cost in a finite horizon. Our analysis is based on a type of finite-horizon occupation measure. We express the distribution of the finite-horizon cost in terms of the occupation measure for each policy, wherein the discount is not needed. For unconstrained and constrained problems, we establish the existence and computation of optimal policies. In particular, we develop a linear program and its dual program for the constrained problem and, moreover, establish the strong duality between the two programs. Finally, we provide two special cases of our results, one of which concerns the discrete-time model, and the other the chance-constrained problem.

Chance-Constrained Programming with Decision-Dependent Uncertainty

We study a class of joint chance-constrained stochastic problems CC-DD with decision-dependent and exogenous uncertainty. A coupling function models the relationship between decision and decision-dependent random variables. We propose reformulations equivalent to the general chance-constrained problem with decision-dependent uncertainty and applicable to any coupling function. We define the properties of coupling functions and explain the importance of properly modeling decision-dependent uncertainty. We then provide the explicit formulation of problem CC-DD in the decision-dependent service uncertainty context and show its versatility and suitability for a wide range of business problems. We design a data-driven algorithmic framework which includes the derivation of convex integer relaxation problems, the use of new multiterm convexification methods, the derivation of tight bounding schemes, and the design of a nonlinear branch-and-bound algorithm featuring a conification method and a new branching rule. Experiments based on real-life data validate the scalability and computational efficiency of the method. To conduct the data-driven analysis, we develop a new medical evacuation chance-constrained model with exogenous and decision-dependent uncertainty which endogenizes the calculation of individual busy probabilities.

An Optimization-Simulation Approach to Chance Constraint Programming

A stochastic programming problem with a number of random parameters in the set of constraints is considered. An iterative optimization-simulation approach is applied as solving method. It consists of two phases: optimization phase, which includes solving a deterministic counterpart of the original chance-constrained problem, and a simulation phase in which the original constraints are checked using Monte Carlo simulation. One iteration corresponds to one scenario. If the decision maker is not satisfied with results, new scenario is generated in which the deterministic values of stochastic parameters are changed in the direction that will provide a more robust solution. Deterministic counterpart in the new scenario is formulated depending on the result of previous iteration. For that purpose, different heuristics are considered. The main goal is to provide a good insight on the optimization problem under uncertainty by performing relatively small number of iterations. General approach and results of the proposed framework will be illustrated on an advertisement placement example.DOI: http://dx.doi.org/10.5755/j01.itc.47.2.18712

Outer-approximation algorithms for nonsmooth convex MINLP problems

In this work, we combine outer-approximation (OA) and bundle method algorithms for dealing with mixed-integer non-linear programming (MINLP) problems with nonsmooth convex objective and constraint functions. As the convergence analysis of OA methods relies strongly on the differentiability of the involved functions, OA algorithms may fail to solve general nonsmooth convex MINLP problems. In order to obtain OA algorithms that are convergent regardless the structure of the convex functions, we solve the underlying OA’s non-linear subproblems by a specialized bundle method that provides necessary information to cut off previously visited (non-optimal) integer points. This property is crucial for proving (finite) convergence of OA algorithms. We illustrate the numerical performance of the given proposal on a class of hybrid robust and chance-constrained problems that involve a random variable with finite support.

A Sequential Algorithm for Solving Nonlinear Optimization Problems with Chance Constraints

An algorithm is presented for solving nonlinear optimization problems with chance constraints, i.e., those in which a constraint involving an uncertain parameter must be satisfied with at least a minimum probability. In particular, the algorithm is designed to solve cardinality-constrained nonlinear optimization problems that arise in sample average approximations of chance-constrained problems, as well as in other applications in which it is only desired to enforce a minimum number of constraints. The algorithm employs a novel exact penalty function which is minimized sequentially by solving quadratic optimization subproblems with linear cardinality constraints. Properties of minimizers of the penalty function in relation to minimizers of the corresponding nonlinear optimization problem are presented, and convergence of the proposed algorithm to stationarity with respect to the penalty function is proved. The effectiveness of the algorithm is demonstrated through numerical experiments with a nonlinear ca...

Managing Reliability and Stability Risks in Forest Harvesting

The timing of forest stands harvesting is an important operational decision in forestry. Major goals of private nonindustrial forest owners are to achieve a steady flow of profits while reaching an overall satisfactory and reliable profit level. These goals are pursued under uncertainties in the growth of trees in different regions and in the prices of wood products. We propose an optimization framework that uses financial risk concepts to capture the above goals and uncertainties, and apply it to a real forestry problem in Finland. Our results demonstrate that the obtained harvesting schedules outperform those obtained without the explicit consideration of the stability and reliability requirements in harvest profits. More generally, our results indicate that the forest owner can improve the profit stability by (i) harvesting a greater number of forest stands early and (ii) harvesting in the first periods of the planning horizon stands that are predominantly composed of slow-growing forests. This research responds to the call for scenario-based approaches that represent well, and in a solvable way, multiple uncertainties in large forestry problems. This study fills in a gap in stochastic programming and can be a cornerstone for subsequent improvements in the solution of combinatorial chance-constrained problems with multirow random technology matrix. The online appendix is available at https://doi.org/10.1287/msom.2017.0626 .

Trading performance for state constraint feasibility in stochastic constrained control: A randomized approach

Constrained control for stochastic linear systems is generally a difficult task due to the possible infeasibility of state constraints. In this paper, we focus on a finite control horizon and propose a design methodology where the constrained control problem is formulated as a chance-constrained optimization problem depending on some parameter. This parameter can be tuned so as to decide the appropriate trade-off between control cost minimization and state constraints satisfaction. An approximate solution is computed via a randomized algorithm. Precise guarantees about its feasibility for the original chance-constrained problem are provided. A numerical example shows the efficacy of the proposed methodology.

Solving Chance-Constrained Optimization Problems with Stochastic Quadratic Inequalities

We propose a new and systematic reformulation and algorithmic approach to solve a complex class of stochastic programming problems involving a joint chance constraint with random technology matrix and stochastic quadratic inequalities. The method is general enough to apply to nonconvex as well as nonseparable quadratic terms. We derive two new reformulations and give sufficient conditions under which the reformulated problem is equivalent. The second reformulation provides a much sparser representation of the feasible set of the chance constraint and offers tremendous computational advantages. This new reformulation method can be used for linear stochastic inequalities and will also significantly improve the solution of such joint chance-constrained problems. We provide general and easily identifiable conditions under which the base reformulations can be linearized. We show that the size of the reformulated problems, in particular, their number of binary variables and quadratic mixed-integer terms, does not grow linearly with the number of scenarios used to represent uncertainty. We propose two new nonlinear branch-and-bound algorithms for the nonconvex quadratic integer reformulations. We present detailed empirical results, comparing the various reformulations and several algorithmic ideas that improve the performance of the mixed-integer nonlinear solver Couenne for solving these problems. Guidelines on how to tune the solver and to select reformulations are presented. The test instances are epidemiology and disaster management facility location models and cover the three types of stochastic quadratic inequalities, namely, product of two decision variables that are both binary, binary and continuous, or both continuous.