Stochastic Quadratic Programming Research Articles

Pulse width modulation control for stochastic Lagrange systems

The pulse width modulation (PWM) control for Lagrange systems driven by servo motors in stochastic environment is researched in this paper. Principles of PWM for stochastic nonlinear systems are proved, a continuous control bounded in probability of stochastic Lagrange system is designed, and the stochastic closed-loop nonlinear programming is modeled in the same form as that in the deterministic case. Sure boundedness of the control is required as a hard constraint, and the stochastic quadratic programming problem is presented for which an analytic solution is constructed and the expression of the optimal duty ratio is obtained. Finally, the main procedures of design and implementation of the stochastic PWM are summarized, and the effectiveness of the proposed scheme is demonstrated by a simulation of articulated manipulator.

A risk-averse stochastic quadratic model with recourse for supporting irrigation water management in uncertain and nonlinear environments

Agricultural water management system are subject to a variety of uncertainties and nonlinearity, which complicate the decision-making process and exaggerate water-shortage risks. Previously, few studies could tackle multiple uncertainties and the associated system risks in nonlinear problems. In this paper, a novel risk-averse optimization model was developed for supporting irrigation water management to mitigate these problems. Based on a risk measurement tool, Conditional Value-at-Risk (CVaR), a CVaR-based interval fuzzy two-stage stochastic quadratic programming (CIFTSQP) was proposed. CIFTSQP could not only provide risk-averse solutions with recourse in response to extreme economic risk quantified as CVaR and water-shortage risk expressed as fuzzy credibility level, but also handle and quantify the nonlinearity in a stochastic program caused by the economy-of-scale effects of facilities. The proposed method has been applied to an irrigation water management case in the northern China. Results suggested that, in response to severe water shortage, sunflowers would be preferred. Meanwhile, groundwater use in Wuyuan County and surface water consumption of Urad Qianqi and Urad Zhongqi should be reduced. Moreover, desired water-allocation strategies with varied risk-aversion levels were generated under different water inflow levels. Results reveal that the highest economic benefits could be achieved when risk aversion degree reaches 30 %, 40 % and 70 % under the low, medium and high inflow levels, respectively. Results reveal that higher risks would bring higher returns under the low and medium inflow levels, and the highest economic benefits could be achieved when risk aversion degree reaches 70 % under the high inflow level.reveal that higher would bring higher returnsunder the low and medium inflow levels, andthe highest economic benefits could be achieved when risk aversion degree reaches 70 % under the high inflow level. The developed CIFTSQP method provided valuable insights into informed irrigation decision-making with respect to economy-of-scale effects, risk aversion and recourse in an agricultural water allocation system. It can also be applied to address other resource allocation problems under uncertainty and risk.

Asymptotic Results of Stochastic Decomposition for Two-Stage Stochastic Quadratic Programming

This paper presents the stochastic decomposition (SD) algorithms for two classes of stochastic programming problems: (1) two-stage stochastic quadratic-linear programming (SQLP) in which a quadrati...

Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic

In this paper, we present a multistage time consistent Expected Conditional Risk Measure for minimizing a linear combination of the expected mean and the expected variance, so-called Expected Mean-Variance. The model is formulated as a multistage stochastic mixed-integer quadratic programming problem combining risk-sensitive cost and scenario analysis approaches. The proposed problem is solved by a matheuristic based on the Branch-and-Fix Coordination method. The multistage scenario cluster primal decomposition framework is extended to deal with large-scale quadratic optimization by means of stage-wise reformulation techniques. A specific case study in risk-sensitive production planning is used to illustrate that a remarkable decrease in the expected variance (risk cost) is obtained. A competitive behavior on the part of our methodology in terms of solution quality and computation time is shown when comparing with plain use of CPLEX in 150 benchmark instances, ranging up to 711,845 constraints and 193,000 binary variables.

Optimizing residential heating and energy storage flexibility for frequency reserves

The increase in intermittent renewable generation motivates participation of also the electricity consumers in maintaining the production–consumption balance of the electrical grid. This paper proposes a distributed price-based optimization scheme for involving a population of consumers in day-ahead procurement of electricity and frequency containment reserves. Energy storage charging and heating of residential houses are planned before the operating day, while taking into account uncertainties in the day-ahead electricity and reserves markets, weather conditions, and realized frequency. The proposed approach is formulated as stochastic quadratic programming problems and coordinated using a parallelized formulation of the alternating direction method of multipliers (ADMM). Chance constraints for the storage state of charge variations under uncertain reserve activation are formulated as second-order cone constraints. The numerical results illustrate the potential benefits of including consumers in the optimization of reserves.

Cooperative Deconflicting Heading Maneuvers Applied to Unmanned Aerial Vehicles in Non-Segregated Airspace

This paper focuses on the conflict detection and resolution (CDR) of unmanned aerial vehicles (UAVs). Firstly, the airspace conflict problem of UAVs is studied and a taxonomy of conflict situation is presented. The multi-UAV conflict is studied in virtue of the graph theory. The CDR problem is casted to a nonlinear optimization problem. Secondly, a two layered optimization algorithm, which combines stochastic parallel gradient descent (SPGD) method and Sequential quadratic programming (SQP) algorithm, is presented to solve the nonlinear optimization problem. Numerical simulations are performed to demonstrate the computational efficiency of this solver. Thirdly, the proposed algorithm is extended to 3-D space. Finally, the proposed algorithm is demonstrated on several scenarios. The results demonstrate that the proposed method outperform the existing algorithms. It can obtain conflict free solutions that would not lead to unnecessary detors.

Solution of Stochastic Quadratic Programming with Imperfect Probability Distribution Using Nelder-Mead Simplex Method

Stochastic quadratic programming with recourse is one of the most important topics in the field of optimization. It is usually assumed that the probability distribution of random variables has complete information, but only part of the information can be obtained in practical situation. In this paper, we propose a stochastic quadratic programming with imperfect probability distribution based on the linear partial information (LPI) theory. A direct optimizing algorithm based on Nelder-Mead simplex method is proposed for solving the problem. Finally, a numerical example is given to demonstrate the efficiency of the algorithm.

A Distributed Interior-Point KKT Solver for Multistage Stochastic Optimization

Multistage stochastic optimization leads to NLPs over scenario trees that become extremely large when many time stages or fine discretizations of the probability space are required. Interior-point methods are well suited for these problems if the arising huge, structured KKT systems can be solved efficiently, for instance, with a large scenario tree but a moderate number of variables per node. For this setting we develop a distributed implementation based on data parallelism in a depth-first distribution of the scenario tree over the processes. Our theoretical analysis predicts very low memory and communication overheads. Detailed computational experiments confirm this prediction and demonstrate the overall performance of the algorithm. We solve multistage stochastic quadratic programs with up to 400 × 106 variables and 8.59 × 109 KKT matrix entries or 136 × 106 variables and 12.6 × 109 entries on a compute cluster with 384 GB RAM. Data are available at https://doi.org/10.1287/ijoc.2017.0748 .

From estimation to optimization via shrinkage

We study a class of quadratic stochastic programs where the distribution of random variables has unknown parameters. A traditional approach is to estimate the parameters using a maximum likelihood estimator (MLE) and to use this as input in the optimization problem. For the unconstrained case, we show that an estimator that shrinks the MLE towards an arbitrary vector yields a uniformly better risk than the MLE. In contrast, when there are constraints, we show that the MLE is admissible.

Single-ratio fractional integer programs with stochastic right-hand sides

ABSTRACTWe present an equivalent value function reformulation for a class of single-ratio Fractional Integer Programs (FIPs) with stochastic right-hand sides and propose a two-phase solution approach. The first phase constructs the value functions of FIPs in both stages. The second phase solves the reformulation using a global branch-and-bound algorithm or a level-set approach. We derive some basic properties of the value functions of FIPs and utilize them in our algorithms. We show that in certain cases our approach can solve instances whose extensive forms have the same order of magnitude as the largest stochastic quadratic integer programs solved in the literature.

A COMPARATIVE STUDY OF STOCHASTIC QUADRATIC PROGRAMMING AND OPTIMAL CONTROL MODEL IN PRODUCTION-INVENTORY SYSTEM WITH STOCHASTIC DEMAND

ABSTRACT. This study compares the optimal control model and stochastic quadratic programming (SQP)model of a production-inventory system.A single product, without shortage in the case of a periodic-reviewpolicy with stochastic demand and deterioration rate as a function of time, is discussed.The items are sub-jected to deterioration via storage and the inventory goal level as a function of production. Demand isrepresented by a stochastic differential equation, converted to a stochastic constraint, and then to a de-terministic constraint. We derive the optimality c onditions of optimal control model and formulate threemodels of SQP. The effect of stochastic demand on the production rate and inventory level is thenillus-trated. Our numerical results appear to suggest that control of inventory level is better in the case of SQP.Furthermore, the total cost is similar in the three models of SQP – despite a difference in production ratesand inventory levels. Keywords : optimal control, production-inventory system, stochastic quadratic programming.

A Quasi-Monte-Carlo-Based Feasible Sequential System of Linear Equations Method for Stochastic Programs with Recourse

A two-stage stochastic quadratic programming problem with inequality constraints is considered. By quasi-Monte-Carlo-based approximations of the objective function and its first derivative, a feasible sequential system of linear equations method is proposed. A new technique to update the active constraint set is suggested. We show that the sequence generated by the proposed algorithm converges globally to a Karush-Kuhn-Tucker (KKT) point of the problem. In particular, the convergence rate is locally superlinear under some additional conditions.

Stochastic quadratic programming problems involving multi-choice parameters

A new solution procedure is developed for stochastic quadratic programming model with some of the right hand side coefficients as multi-choice parameter and rest of the coefficients are considered as random variables. In the present paper, randomness is measured by exponential random variables and multi-choiceness is characterized by interpolating polynomial approach namely Lagrange interpolating polynomial. In order to solve the model, first convert the proposed model into an equivalent deterministic model; then remove the multi-choice parameter by using Lagrange interpolating polynomial. Finally, a suitable numerical example is presented to illustrate the proposed solution procedure.

Continuity and stability of two-stage stochastic programs with quadratic continuous recourse

For two-stage stochastic programs with quadratic continuous recoursewhere all the coefficients in the objective function and theright-hand side vector in the second-stage constraints varysimultaneously, we firstly show the locally Lipschtiz continuity ofthe optimal value function of the recourse problem, then undersuitable probability metric, we derive the joint Lipschitzcontinuity of the expected optimal value function with respect tothe first-stage variables and the probability distribution.Furthermore, we establish the qualitative and quantitative stabilityresults of the optimal value function and the optimal solution setwith respect to the Fortet-Mourier probability metric, when theunderlying probability distribution is perturbed. Finally, we showthe exponential convergence rate of the optimal value sequence whenwe solve two-stage quadratic stochastic programs by the sampleaverage approximation method.

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.

Development of a fuzzy-queue-based stochastic quadratic program for water resources planning

In this study, a fuzzy-queue (FQ)-based inexact stochastic quadratic programming (SQP) method is developed through coupling FQ technique with inexact SQP. FQ-SQP improves upon the existing stochastic programming methods by considering the effects of queuing phenomenon during the water resources allocation process. FQ-SQP cannot only handle uncertainties expressed as interval values, random variables, and fuzzy sets, but also tackle nonlinearity in the objective function; more importantly, it can reflect the effects of FQ on water resources allocation and system benefit. The FQ-SQP model is applied to a case study of planning water resources management, where FM/FM/1 (fuzzy exponential interarrival time, fuzzy exponential service time, and one server) queue is incorporated within the SQP modeling framework. Based on α-cut analysis technique, interval solutions with fuzzy arrival and service rates have been generated, which result in different water resources allocation patterns as well as changed waiting water amounts and system benefits. Results indicate that consideration of queuing problem impacts on water resources allocation can provide more useful information for decision makers and gain in-depth insights into the effects of queuing problems for water resources allocation.

A Polynomial-Time Solution Scheme for Quadratic Stochastic Programs

We consider quadratic stochastic programs with random recourse—a class of problems which is perceived to be computationally demanding. Instead of using mainstream scenario tree-based techniques, we reduce computational complexity by restricting the space of recourse decisions to those linear and quadratic in the observations, thereby obtaining an upper bound on the original problem. To estimate the loss of accuracy of this approach, we further derive a lower bound by dualizing the original problem and solving it in linear and quadratic recourse decisions. By employing robust optimization techniques, we show that both bounding problems may be approximated by tractable conic programs.

Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.

A new optimal electricity market bid model solved through perspective cuts

On current electricity markets the electrical utilities are faced with very sophisticated decision making problems under uncertainty. Moreover, when focusing in the short-term management, generation companies must include some medium-term products that directly influence their short-term strategies. In this work, the bilateral and physical futures contracts are included into the day-ahead market bid following MIBEL rules and a stochastic quadratic mixed-integer programming model is presented. The complexity of this stochastic programming problem makes unpractical the resolution of large-scale instances with general-purpose optimization codes. Therefore, in order to gain efficiency, a polyhedral outer approximation of the quadratic objective function obtained by means of perspective cuts (PC) is proposed. A set of instances of the problem has been defined with real data and solved with the PC methodology. The numerical results obtained show the efficiency of this methodology compared with standard mixed quadratic optimization solvers.

A stochastic programming model for the thermal optimal day-ahead bid problem with physical futures contracts

The reorganization of the electricity industry in Spain completed a new step with the start-up of the Derivatives Market. One main characteristic of MIBEL's Derivatives Market is the existence of physical futures contracts; they imply the obligation to physically settle the energy. The market regulation establishes the mechanism for including those physical futures in the day-ahead bidding of the generation companies. The goal of this work is to optimize coordination between physical futures contracts and the day-ahead bidding which follow this regulation. We propose a stochastic quadratic mixed-integer programming model which maximizes the expected profits, taking into account futures contracts settlement. The model gives the simultaneous optimization for the Day-Ahead Market bidding strategy and power planning production (unit commitment) for the thermal units of a price-taker generation company. The uncertainty of the Day-Ahead Market price is included in the stochastic model through a set of scenarios. Implementation details and some first computational experiences for small real cases are presented.