Multiple Inequality Constraints Research Articles

Quantum Annealing with Inequality Constraints: The Set Cover Problem

AbstractQuantum annealing is a promising method for solving hard optimization problems by transforming them into quadratic unconstrained binary optimization (QUBO) problems. However, when constraints are involved, particularly multiple inequality constraints, incorporating them into the objective function poses challenges. In this paper, the authors present two novel approaches for solving problems with multiple inequality constraints on a quantum annealer and apply them to the set cover problem (SCP). The first approach uses the augmented Lagrangian method to represent the constraints, while the second approach employs a higher‐order binary optimization (HUBO) formulation. The experiments show that both approaches outperform the standard approach for solving the SCP on the D‐Wave Advantage quantum annealer. The HUBO formulation performs slightly better than the augmented Lagrangian method in solving the SCP, but its scalability in terms of embeddability in the quantum chip is worse. The results demonstrate that the proposed augmented Lagrangian and HUBO methods can successfully implement a large number of inequality constraints, making them applicable to a broad range of constrained problems beyond the SCP.

Constrained Information Design

We provide tools to analyze information design problems subject to constraints. We do so by extending an insight by Le Treust and Tomala to the case of multiple inequality and equality constraints. Namely, that an information design problem subject to constraints can be represented as an unconstrained information design problem with additional states, one for each constraint. Thus, without loss of generality, optimal solutions induce as many posteriors as the number of states and constraints. We provide results that refine this upper bound. Furthermore, we provide conditions under which there is no duality gap in constrained information design, thus validating a Lagrangian approach. We illustrate our results with applications to mechanism design with limited commitment and persuasion of a privately informed receiver. Funding: L. Doval acknowledges the support of the National Science Foundation through [Grant SES-2131706]. V. Skreta acknowledges the support from the National Science Foundation through [Grant SES-1851729] and from the European Research Council (ERC) through consolidator [Grant 682417]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.1346 .

Optimal Path Following Control With Efficient Computation for Snake Robots Subject to Multiple Constraints and Unknown Frictions

This letter proposes a real-time optimal robust path following control scheme for planar snake robots without sideslip constraints using model predictive control (MPC). One of the features is that a linear double-integrator model rather than the complex dynamic model of snake robots is used for the MPC design to improve calculation efficiency. Moreover, in addition to constraints on joint angles, velocity, and acceleration for security, constraints on the joint offset and velocity are also considered to keep the snake robot moving forward efficiently. And the multiple inequality constraints are handled by a novel constraints translator. Furthermore, to achieve robust path following control subject to unknown and varied friction forces and disturbances, two reduced-order structure-improved extended state observers are designed to avoid complex environmental modeling. Extensive comparative experiments were conducted to verify the constraints design and the effectiveness of the proposed control scheme.

Dynamic Balancing of Humanoid Robot with Proprioceptive Actuation: Systematic Design of Algorithm, Software, and Hardware.

For humanoid robots, maintaining a dynamic balance against uncertain disturbance is crucial, and this function can be achieved by coordinating the whole body to perform multiple tasks simultaneously. Researchers generally accept hierarchical whole-body control (WBC) to address this function. Although experts can build feasible hierarchies using prior knowledge, real-time WBC is still challenging because it often requires a quadratic program with multiple inequality constraints. In addition, the torque tracking performance of the WBC algorithm will be affected by uncertain factors such as joint friction for a large transmission ratio proprioceptive-actuated robot. Therefore, the balance control of physical robots requires a systematic solution. In this study, a robot control system with high computing power and real-time communication ability, UBTMaster, is implemented to achieve a reduced WBC in real time. Based on these, a whole-body control scheme based on task priority for the dynamic balance of humanoid robots is implemented. After realizing the joint friction model identification, finally, a variety of balancing scenarios are tested on the Walker3 humanoid robot driven by the proprioceptive actuators to verify the effectiveness of the proposed scheme. The Walker3 robot exhibits excellent balance when multiple external disturbances occur simultaneously. For example, the two feet of the robot are subjected to tilt and displacement perturbations, respectively, while the torso is subjected to external shocks simultaneously. The experimental results show that the dynamic balance of the robot under multiple external disturbances can be achieved by using strictly hierarchical real-time WBC with a systematic design.

En-Route Charging Strategy for Wirelessly Charged Electric Bus Considering Time-of-Use Price

To mitigate the range anxiety problem of electric bus system, wireless power transfer is regarded as one of the emerging technologies for long-term range extension. Previous studies have discussed the optimization problem of the power track deployment. However, the en-route charging strategy also significantly influences the operation cost besides the power track, which is yet to be investigated sufficiently. To fill this gap, a new wireless charging model for optimizing the energy cost is proposed. In particular, the cost of battery and the time-of-use electricity price are taken into account. Firstly, a microscopic power consumption model considering passenger flows and automobile dynamics is developed to estimate the charging cost. Then, a relaxation approach based on penalty function and grey wolf optimization (GWO) algorithm is developed to solve the non-deterministic polynomial-hard (NP-hard) problem with complex multidimensional variables and multiple inequality constraints. And the performance of the proposed charging strategy is verified in a real-world bus line via numerical simulation. A sensitivity analysis is conducted to quantify the marginal impact of the unit cost of battery capacity on the total energy cost. Finally, the computational performance of the proposed algorithm (GWO) is validated by comparing other outstanding methods such as genetic algorithm (GA), particle swarm optimization (PSO) and CPLEX solvers.

Generalized Optimality Criteria Method for Topology Optimization

In this article, a generalized optimality criteria method is proposed for topology optimization with arbitrary objective function and multiple inequality constraints. This algorithm uses sensitivity information to update both the Lagrange multipliers and design variables. Different from the conventional optimality criteria method, the proposed method does not satisfy constraints at every iteration. Rather, it improves the Lagrange multipliers and design variables such that the optimality criteria are satisfied upon convergence. The main advantages of the proposed method are its capability of handling multiple constraints and computational efficiency. In numerical examples, the proposed method was found to be more than 100 times faster than the optimality criteria method and more than 1000 times faster than the method of moving asymptotes.

Optimal control of vibrations of two nonlinear Gao beams connected with a joint

This paper is concerned with optimal control of vibrations of two uniform elastic or viscoelastic nonlinear Gao beams that are connected with a joint. The dynamic contact is modelled with the Signorini non-penetration or unilateral conditions in which the stops are assumed to be perfectly rigid. By the Dubovitskii and Milyutin functional analytical approach, we derive the Pontryagin maximum principle for the optimal control problem with multiple equality and multiple inequality constraints in fixed final horizon case. A remark is made on how to use the obtained results in the case of a quadratic cost functional.

Maximum principle for optimal control of vibrations of a dynamic Gao beam in contact with a rigid foundation

ABSTRACTIn our preceding paper, we studied an optimal control problem of vibrations of a dynamic Gao beam in contact with a reactive foundation and derived the Pontryagin maximum principle for the controlled system in fixed final horizon case. As a follow-up, in this paper, we focus on the investigation of the Gao beam that may come in contact with a rigid foundation underneath it. In this case, the nonlinear viscoelastic beam equation is equipped with the Signorini condition. By the Dubovitskii and Milyutin functional analytical approach, we investigate the new optimal control problem with multiple inequality constraints and present further original results of current interests.

Lower Bounds on Integrated Risk, Subject to Inequality Constraints

SummaryStatistical decision theory can sometimes be used to find, via a least favourable prior distribution, a statistical procedure that attains the minimax risk. This theory also provides, using an ‘unfavourable prior distribution’, a very useful lower bound on the minimax risk. In the late 1980s, Kempthorne showed how, using a least favourable prior distribution, a specified integrated risk can sometimes be minimised, subject to an inequality constraint on a different risk. Specifically, he was concerned with the solution of a minimax‐Bayes compromise problem (‘compromise decision theory’). Using an unfavourable prior distribution, Kabaila & Tuck (), provided a very useful lower bound on an integrated risk, subject to an inequality constraint on a different risk. We extend this result to the case of multiple inequality constraints on specified risk functions and integrated risks. We also describe a new and very effective method for the computation of an unfavourable prior distribution that leads to a very useful lower bound. This method is simply to maximize the lower bound directly with respect to the unfavourable prior distribution. Not only does this method result in a relatively tight lower bound, it is also fast because it avoids the repeated computation of the global maximum of a function with multiple local maxima. The advantages of this computational method are illustrated using the problems of bounding the performance of a point estimator of (i) the multivariate normal mean and (ii) the univariate normal mean.

A Bayesian approach to constrained single- and multi-objective optimization

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria---the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions---as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization.

Resource Allocation for Energy Efficiency Optimization in Heterogeneous Networks

Heterogeneous network (HetNet) deployment is considered a de facto solution for meeting the ever increasing mobile traffic demand. However, excessive power usage in such networks is a critical issue, particularly for mobile operators. Characterizing the fundamental energy efficiency (EE) performance of HetNets is therefore important for the design of green wireless systems. In this paper, we address the EE optimization problem for downlink two-tier HetNets comprised of a single macro-cell and multiple pico-cells. Considering a heterogeneous real-time and non-real-time traffic, transmit beamforming design and power allocation policies are jointly considered in order to optimize the system energy efficiency. The EE resource allocation problem under consideration is a mixed combinatorial and non-convex optimization problem, which is extremely difficult to solve. In order to reduce the computational complexity, we decompose the original problem with multiple inequality constraints into multiple optimization problems with single inequality constraint. For the latter problem, a two-layer resource allocation algorithm is proposed based on the quasiconcavity property of EE. Simulation results confirm the theoretical findings and demonstrate that the proposed resource allocation algorithm can efficiently approach the optimal EE.

Hidden Markov model parameters estimation with independent multiple observations and inequality constraints

In this study, the authors focus on hidden Markov model (HMM) parameters estimation with independent multiple observations and non-linear inequality constraints. The parameters estimation process is divided into four steps: initialisation, parameters pre-estimation, parameters re-estimation and termination. The pre-estimation results are used to approximate non-linear inequality constraints to linear inequality constraints. In parameters re-estimation step, the active-set optimisation is combined with the expectation maximisation (EM) algorithm in M-step and the active set-based EM algorithm is proposed to re-estimate HMM parameters when inequality constraints are not satisfied in pre-estimation. An auxiliary function is devised for reconstructing the optimisation objective function and the convergence of the proposed algorithm is also demonstrated. Simulation results indicate that the proposed algorithm provides better performance by modifying the random error of observation data appropriately and it is powerful for industry process fault diagnosis.

Stochastic approximation algorithms for constrained optimization via simulation

We develop four algorithms for simulation-based optimization under multiple inequality constraints. Both the cost and the constraint functions are considered to be long-run averages of certain state-dependent single-stage functions. We pose the problem in the simulation optimization framework by using the Lagrange multiplier method. Two of our algorithms estimate only the gradient of the Lagrangian, while the other two estimate both the gradient and the Hessian of it. In the process, we also develop various new estimators for the gradient and Hessian. All our algorithms use two simulations each. Two of these algorithms are based on the smoothed functional (SF) technique, while the other two are based on the simultaneous perturbation stochastic approximation (SPSA) method. We prove the convergence of our algorithms and show numerical experiments on a setting involving an open Jackson network. The Newton-based SF algorithm is seen to show the best overall performance.

An actor–critic algorithm with function approximation for discounted cost constrained Markov decision processes

We develop in this article the first actor–critic reinforcement learning algorithm with function approximation for a problem of control under multiple inequality constraints. We consider the infinite horizon discounted cost framework in which both the objective and the constraint functions are suitable expected policy-dependent discounted sums of certain sample path functions. We apply the Lagrange multiplier method to handle the inequality constraints. Our algorithm makes use of multi-timescale stochastic approximation and incorporates a temporal difference (TD) critic and an actor that makes a gradient search in the space of policy parameters using efficient simultaneous perturbation stochastic approximation (SPSA) gradient estimates. We prove the asymptotic almost sure convergence of our algorithm to a locally optimal policy.

Testing for Complementarity and Substitutability in the Case of Multiple Practices

Recent empirical studies of firm-level performance have been concerned with establishing potential complementarity between more than two organizational practices. These papers have drawn conclusions on the basis of potentially biased estimates of pair-wise interaction effects between such practices. In this paper we develop a consistent testing framework based on multiple inequality constraints that derives from the definition of (strict) supermodularity as suggested by Athey and Stern (1998). Monte Carlo results show that the multiple restrictions test is superior for performance models with high explanatory power. If practices explain only a minor part of organizational performance no test is able to identify complementarity or substitutability in a satisfactory manner.

Analytic Solutions for Equal Mass Four-Craft Static Coulomb Formation

Analytic constant charge solutions are investigated for square planar and 3-D tetrahedron four-craft static Coulomb formations. The solutions are formulated in terms of the formation geometry and attitude. In contrast to the two- and three-spacecraft Coulomb formations, a four-spacecraft formation has new equality and multiple inequality constraints that need to be satisfied for the individual spacecraft charges to be both unique and real. Unique charge relative equilibriums are important to reduce the overall power requirement of the spacecraft charge emision. A spacecraft must not only satisfy three inequality three-craft constraints to yield a real charge solution, but it must also satisfy two additional equality constraints to ensure that the spacecraft charges are unique. Further, a method is presented to reduce the number of equality constraints arising due to the dynamics of a four-spacecraft formation. The unique and real spacecraft charges are determined as a function of the orientation of the square formation in a given principal orbit plane. For the 3-D tetrahedron formation scenario there is only a unique set of charged products. The implementability constraints are numerically evaluated to show that only trivial equal-mass tetrahedron formations are possible where one craft is on the along-track axis with zero charge.

Successive equimarginal approach for optimal design of a pump and treat system

An economic concept‐based optimization method is developed for groundwater remediation design. Design of a pump and treat (P&T) system is viewed as a resource allocation problem constrained by specified cleanup criteria. An optimal allocation of resources requires that the equimarginal principle, a fundamental economic principle, must hold. The proposed method is named successive equimarginal approach (SEA), which continuously shifts a pumping rate from a less effective well to a more effective one until equal marginal productivity for all units is reached. Through the successive process, the solution evenly approaches the multiple inequality constraints that represent the specified cleanup criteria in space and in time. The goal is to design an equal protection system so that the distributed contaminant plumes can be equally contained without bypass and overprotection is minimized. SEA is a hybrid of the gradient‐based method and the deterministic heuristics‐based method, which allows flexibility in dealing with multiple inequality constraints without using a penalty function and in balancing computational efficiency with robustness. This method was applied to design a large‐scale P&T system for containment of multiple plumes at the former Blaine Naval Ammunition Depot (NAD) site, near Hastings, Nebraska. To evaluate this method, the SEA results were also compared with those using genetic algorithms.

Proper efficiency and duality for a class of multiobjective fractional variational problems containing arbitrary norms

Necessary and sufficient conditions are established for properly efficient solutions of a class of nonsmooth nonconvex variational problems with multiple fractional objective functions and nonlinear inequality constraints. Based on these proper efficiency criteria. two multiobjective dual problems are constructed and appropriate duality theorems are proved. These proper efficiency and duality results also contain as special cases similar rcsults fer constrained variational problems with multiplei fractional. and conventional objective functions, which are particular cases of the main variational problem considered in this paper

Mixed L2and L∞problems by weight selection in quadratic optimal control

In an attempt to achieve more realistic control objectives, the weighting matrices in the standard LQ1 problem are usually chosen by the designer in an ad hoc manner. This paper shows several optimal control design problems that minimize a quadratic function of the control vector subject to multiple inequality constraints on the output L 2 norms, L ∞ norms, covariance matrix, and the maximum singular value of the output covariance matrix. The solutions of all four of these problems reduce to standard LQI control problems with different choices of weights. This paper shows how to construct these different weights. The practical significance of these results is that many robustness properties relate directly to these four entities. Hence the given control design algorithm delivers a specified degree of robustness to both parameter errors and disturbances. The results are presented in the deterministic terms of the linear quadratic impulse (LQI)for continuous and discrete systems problem rather than the stoc...

An Exact Test for Multiple Inequality and Equality Constraints in the Linear Regression Model

Abstract In this article we consider the linear regression model y = Xβ + e, where e is N(0, σ2I). In this context we derive exact tests of the form H: Rβ ≥ r versus K: β ∈ RK for the case in which θ2 is unknown. We extend these results to consider hypothesis tests of the form H: R1β ≥ r1 and R2β = r2 versus K: (β ∈ RK . For each of these hypotheses tests we derive several equivalent forms of the test statistics using the duality theory of the quadratic programming. For both tests we derive their exact distribution as a weighted sum of Snedecor's F distributions normalized by the numerator degrees of freedom of each F distribution of the sum. A methodology for computing critical values as well as probability values for the tests is discussed. The relationship between this testing framework and the multivariate one-sided hypothesis testing literature is also discussed. In this context we show that for any size of the hypothesis test H: λ = 0 versus K: β ∈ RK the test statistic and critical value obtained a...