Redundant Constraints Research Articles

A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem

The minimum cut problem, MC, and the special case of the vertex separator problem, consists in partitioning the set of nodes of a graph G into k subsets of given sizes in order to minimize the number of edges cut after removing the k-th set. Previous work on approximate solutions uses, in increasing strength and expense: eigenvalue, semidefinite programming, SDP, and doubly nonnegative, DNN, bounding techniques. In this paper, we derive strengthened SDP and DNN relaxations, and we propose a scalable algorithmic approach for efficiently evaluating, theoretically verifiable, both upper and lower bounds. Our stronger relaxations are based on a new gangster set, and we demonstrate how facial reduction, FR, fits in well to allow for regularized relaxations. Moreover, the FR appears to be perfectly well suited for a natural splitting of variables, and thus for the application of splitting methods. Here, we adopt the strictly contractive Peaceman-Rachford splitting method, sPRSM. Further, we bring useful redundant constraints back into the subproblems, and show empirically that this accelerates sPRSM.In addition, we employ new strategies for obtaining lower bounds and upper bounds of the optimal value of MC from approximate iterates of the sPRSM thus aiding in early termination of the algorithm. We compare our approach with others in the literature on random datasets and vertex separator problems. This illustrates the efficiency and robustness of our proposed method.

On removing potential redundant constraints for SVOR learning

As an extension of support vector machine in ordinal regression problem, support vector ordinal regression (SVOR) finds (r−1) parallel hyperplanes by solving a quadratic programming which has n(r−1) constraints. Here, n and r represent the number of training sample and ranks, respectively. Therefore, it would costs much more time to train SVOR than SVC or SVR. Fortunately, the solution of SVOR is only decided by minor constraints which are associated with non-zero Lagrange multipliers. Other constraints with zero Lagrange multipliers have no influence on the solution. Because a training sample is associated with (r−1) constraints, retaining potential support vector may still induce that many redundant constraints are reserved. In this paper, we try to remove these potential redundant constraints for SVOR learning. For the jth parallel hyperplane, the potential constraints with non-zero Lagrange multipliers are associated with the samples near the jth parallel hyperplane. These samples can be identified by a chain near the jth parallel hyperplane. Then, other constraints for the jth parallel hyperplane can be discarded before learning. The number of the constraints can be reduced to less than 17 percent of the original in our experiments. Additionally, it only executes once to find potential critical constraints. Obviously, it is easy to tune parameters after removing potential redundant constraints. The experimental results on several datasets demonstrate that SVOR becomes much faster after removing potential redundant constraints and the performance does not degrade seriously.

Gauss optimization method for the dynamics of unilateral contact of rigid multibody systems

The discontinuous dynamical problem of multi-point contact and collision in multi-body system has always been a hot and difficult issue in this field. Based on the Gauss’ principle of least constraint, a unified optimization model for multibody system dynamics with multi-point contact and collision is established. The paper presents the study of the numerical solution scheme, in which particle swarm optimization method is used to deal with the corresponding optimization model. The article also presents the comparison of the Gauss optimization method (GOM) and the hybrid linear complementarity method (i.e. combining differential algebraic equations (DAEs) and linear complementarity problems (LCP)), commonly used to solve the dynamic contact problem of multibody systems with bilateral constraints. The results illustrate that, the GOM has the same advantage of dynamical modelling with LCP and when the redundant constraint exists, the GOM always has a unique solution and so no additional processing is needed, whereas the corresponding DAE-LCP method may have singular cases with multiple solutions or no solutions. Using numerical examples, the GOM is verified to effectively solve the dynamics of multibody systems with redundant unilateral and bilateral constraints without additional redundancy processing. The GOM can also be applied to the optimal control of systems in the future and combined with the parameter optimization of systems to handle dynamic problems. The work given provides the dynamics and control of the complex system with a new train of thought and method. The paper establishes a unified optimization model and the corresponding numerical solution scheme for multibody system dynamics with multi-point contact and collision based on Gauss’ principle of least constraint. Comparisons between Gauss optimization method (GOM) and the hybrid linear complementarity method (HLCM) are carried out through theoretical analysis and numerical examples, which show that GOM has the same advantage of dynamical modelling as HLCM and when redundant constraints exist, GOM has unique solutions without additional processes, whereas HLCM may have singular cases with multiple or no solutions.

Coalitional Game Theory Based Value Sharing in Energy Communities

This paper presents a coalitional game for value sharing in energy communities (ECs). It is proved that the game is super-additive, and the grand coalition effectively increases the global payoff. It is also proved that the model is balanced and thus, it has a nonempty core. This means there always exists at least one value sharing mechanism that makes the grand coalition stable. Therefore, prosumers will always achieve lower bills if they join to form larger ECs. A counterexample is presented to demonstrate that the game is not convex and value sharing based on Shapley values does not necessarily ensure the stability of the coalition. To find a stabilizing value sharing mechanism that belongs to the core of the game, the worst-case excess minimization concept is applied. In this concept, however, size of the optimization problem increases exponentially with respect to the number of members in EC. To make the problem computationally tractable, the idea of clustering members based on their generation/load profiles and considering the same profile and share for members in the same cluster is proposed here. K-means algorithm is used for clustering prosumers’ profiles. This way, the problem would have several redundant constraints that can be removed. The redundant constraints are identified and removed via the generalized Llewellyn’s rules. Finally, value sharing in an apartment building in the southern part of Finland in the metropolitan area is studied to demonstrate effectiveness of the method.

New redundant constraints for Klee-Minty problem

Interior-point method (IPM) is a class of methods which has polynomial time for iteration complexity in solving linear optimisation problems. The iteration complexity upper bound of IPM can be stat...

New redundant constraints for Klee-Minty problem

New redundant constraints for Klee-Minty problem

Application of Gauss principle of least constraint in multibody systems with redundant constraints

Redundancy in the constrained mechanical systems often occurs in complex multibody mechanic systems in the existence of excessive constraints and singular positions due to system motion. In this work, Gauss principle of least constraint (GPLC) is applied to solve the dynamic motion of system with redundant constraints without changing the physics of system. Furthermore, the particle swarm optimization method is used to handle the minimization optimization problem. Eventually, the effectiveness of GPLC is validated through the dynamic modelling and simulation of two numerical examples (a planar four-bar mechanism and a spatial parallelogram mechanism). The simulation results are analyzed and compared with those obtained from Udwaia-Phohomsiri formulation and augmented Lagrangian formulation, in terms of constraint violation, computational efficiency and variation of the mechanical energy. From the viewpoint of computational efficiency and accuracy, GPLC can be regarded as a practical real-time simulation method for multibody systems with redundant constraints.

Linear Programming and Its Application Techniques in Optimizing Portfolio Selection of a Firm

Optimization techniques have been used in this paper to obtain an optimal investment in a selected portfolio that gives maximum returns with minimal inputs based on the secondary data supplied by a particular firm that is examined. Sensitivity analysis is done to ascertain the robustness of the resulting model towards the changes in input parameters to determine a redundant constraint using linear programming. The challenge of determining the available funds and allocating each component of the portfolio to maximize returns and minimize inputs by portfolio holders and managers who are the major decision-makers in allocating their resources cannot be quantified. This optimization technique is used to obtain an optimal investment portfolio including financial risks of a firm with disposable of $15,000,000.00 invested in crude oil, mortgage securities, cash crop, certificate of deposit, fixed deposit, treasury bills, and construction loans. The model is a single-objective model that maximizes the return on the portfolio as the interests on the original data reduces by 5%; then, the return on investments also reduced by almost 15%, with the quantum of money on treasury bills and construction loans posing a significant reduction for the maximum return. The investment in the other options saw a slight decrease. Also, as the interest rates of the original data increase by 5%, the return on investments also grows by almost 17% while the quantum of money on the treasury bills and construction loans increases, and the quantum of money on the other options experienced a decrease except for mortgage securities which recorded a slight increase.

Structural performance envelopes in load space

ABSTRACTVisualising the loads that a structure can tolerate provides a key insight into the structural design process, especially for materials and structures that are governed by complex failure criteria. This paper proposes a general method for efficient construction of performance envelopes in load space, and demonstrates the approach with two examples. The performance envelope identifies all possible failure modes, all the redundant and non-redundant structural constraints, and the limiting failure mode in a particular direction in load space. Once the envelope has been constructed, the structural reserve factors can be calculated extremely quickly. In design such envelopes are most useful for structural analysis processes which involve a very large number of load cases, and where the cost of constructing an envelope for a given feature is relatively modest.

A New Method for Type Synthesis of 2R1T and 2T1R 3-DOF Redundant Actuated Parallel Mechanisms with Closed Loop Units

The current type synthesis of the redundant actuated parallel mechanisms is adding active-actuated kinematic branches on the basis of the traditional parallel mechanisms, or using screw theory to perform multiple getting intersection and union to complete type synthesis. The number of redundant parallel mechanisms obtained by these two methods is limited. In this paper, based on Grassmann line geometry and Atlas method, a novel and effective method for type synthesis of redundant actuated parallel mechanisms (PMs) with closed-loop units is proposed. Firstly, the degree of freedom (DOF) and constraint line graph of the moving platform are determined successively, and redundant lines are added in constraint line graph to obtain the redundant constraint line graph and their equivalent line graph, and a branch constraint allocation scheme is formulated based on the allocation criteria. Secondly, a scheme is selected and redundant lines are added in the branch chains DOF graph to construct the redundant actuated branch chains with closed-loop units. Finally, the branch chains that meet the requirements of branch chains configuration criteria and F&C (degree of freedom & constraint) line graph are assembled. In this paper, two types of 2 rotational and 1 translational (2R1T) redundant actuated parallel mechanisms and one type of 2 translational and 1 rotational (2T1R) redundant actuated parallel mechanisms with few branches and closed-loop units were taken as examples, and 238, 92 and 15 new configurations were synthesized. All the mechanisms contain closed-loop units, and the mechanisms and the actuators both have good symmetry. Therefore, all the mechanisms have excellent comprehensive performance, in which the two rotational DOFs of the moving platform of 2R1T redundant actuated parallel mechanism can be independently controlled. The instantaneous analysis shows that all mechanisms are not instantaneous, which proves the feasibility and practicability of the method.

Единая теория структуры, синтеза и анализа многозвенных механических систем с геометрическими, гибкими и динамическими связями звеньев. Часть 2. Предельные теоремы и области существования

This paper examines a family of limiting structural and topological theorems complemented by two well-known Gruebler theorems that can be used for determining the limiting areas of existence of possible multibody mechanisms. All the multiloop linkage, cam and gear mechanisms synthesized using the proposed unified theory have the optimal structure without redundant constraints and uncontrolled movements.

On neighbourhood singleton-style consistencies for qualitative spatial and temporal reasoning

Given a qualitative constraint network (QCN), a singleton-style consistency focuses on each base relation (atom) of a constraint separately, rather than the entire constraint altogether. This local consistency is essential for tackling fundamental reasoning problems associated with QCNs, such as minimal labeling, but can suffer from redundant constraint checks, especially when checks occur far from where the pruning usually takes place. In this paper, we propose singleton-style consistencies that are applied just on the neighbourhood of a singleton-checked constraint instead of the whole network. We make a theoretical comparison with existing consistencies and consequently prove some properties of the new ones. Further, we propose algorithms to enforce our consistencies, as well as parsimonious variants thereof, that are more efficient in practice than the state of the art. An experimental evaluation with random and structured QCNs of Allen's Interval Algebra in the phase transition region demonstrates the potential of our approach.

ROSE: Robustly Safe Charging for Wireless Power Transfer

One critical issue for wireless power transfer is to avoid human health impairments caused by electromagnetic radiation (EMR) exposure. The existing studies mainly focus on scheduling wireless chargers so that (expected) EMR at any point in the area does not exceed a threshold <inline-formula><tex-math notation="LaTeX">$R_t$</tex-math></inline-formula> . Nevertheless, they overlook the EMR jitter that leads to exceeding of <inline-formula><tex-math notation="LaTeX">$R_t$</tex-math></inline-formula> even if the expected EMR is no more than <inline-formula><tex-math notation="LaTeX">$R_t$</tex-math></inline-formula> . This paper studies the fundamental problem of <u>RO</u> bustly <u>S</u> af <u>E</u> charging for wireless power transfer (ROSE), that is, scheduling the power of chargers so that the charging utility for all rechargeable devices is maximized while the probability that EMR anywhere does not exceed <inline-formula><tex-math notation="LaTeX">$R_t$</tex-math></inline-formula> is no less than a given confidence. We first build our empirical probabilistic charging model and EMR model. Then, we present EMR approximation and area discretization techniques to formulate ROSE into a Second-Order Cone Program. After that, we propose the first redundant second-order cone constraints reduction algorithm to reduce the computational cost, and therefore obtain a <inline-formula><tex-math notation="LaTeX">$(1-\epsilon)$</tex-math></inline-formula> -approximation centralized algorithm. Further, we propose a <inline-formula><tex-math notation="LaTeX">$(1-\epsilon)$</tex-math></inline-formula> -approximation fully distributed algorithm scalable with network size for ROSE. We conduct both simulation and field experiments, and the results show that our algorithms can outperform comparison algorithms by 480.19 percent.

Learning to Solve Large-Scale Security-Constrained Unit Commitment Problems

Security-constrained unit commitment (SCUC) is a fundamental problem in power systems and electricity markets. In practical settings, SCUC is repeatedly solved via mixed-integer linear programming (MIP), sometimes multiple times per day, with only minor changes in input data. In this work, we propose a number of machine learning techniques to effectively extract information from previously solved instances in order to significantly improve the computational performance of MIP solvers when solving similar instances in the future. Based on statistical data, we predict redundant constraints in the formulation, good initial feasible solutions, and affine subspaces where the optimal solution is likely to lie, leading to a significant reduction in problem size. Computational results on a diverse set of realistic and large-scale instances show that using the proposed techniques, SCUC can be solved on average 4.3 times faster with optimality guarantees and 10.2 times faster without optimality guarantees, with no observed reduction in solution quality. Out-of-distribution experiments provide evidence that the method is somewhat robust against data-set shift.Summary of Contribution. The paper describes a novel computational method, based on a combination of mixed-integer linear programming (MILP) and machine learning (ML), to solve a challenging and fundamental optimization problem in the energy sector. The method advances the state-of-the-art, not only for this particular problem, but also, more generally, in solving discrete optimization problems via ML. We expect that the techniques presented can be readily used by practitioners in the energy sector and adapted, by researchers in other fields, to other challenging operations research problems that are solved routinely.

Cable Angle and Minimum Resultant Force Response Analysis of Lower Limb Traction Device for Rehabilitation Robot With Interval Parameters

Abstract This paper proposes a hybrid uncertainty analysis method (HUAM) based on the first-order interval perturbation method (FIPM) and Monte Carlo method (MCM) for minimum resultant force response analysis of the lower limb traction device (LLTD) of a hybrid-driven parallel waist rehabilitation robot (HDPWRR) with interval parameters. Based on the analysis of cable angles by using the interval algorithm, the problem of non-uniqueness of the force solution in redundant constraint mechanisms is solved. The force response domain prediction with interval parameters on rehabilitation patients is estimated by using the HUAM which combining the first-order interval perturbation technique with direct Monte Carlo method in different stages, and it reduces the calculation amount. First, the kinematic and static models of the LLTD with deterministic information are established according to its work principle. Then, the interval matrices with interval parameters are calculated by using the FIPM and the response of cable angles is combined with the static model. Third, by numerical examples, the accuracy and efficiency of the HUAM for solving the force response domain problem with interval parameters are verified. The bounds of cable angle response domain of the interval LLTD model are determined. Finally, the minimum resultant force response domain prediction with interval parameters on rehabilitation patients is estimated by combining the FIPM and MCM.

A regularization method for solving the redundant problems in multibody dynamic system

Redundant constraints may cause great difficulties during the numerical simulation, when solving the differential algebraic equations (DAEs). The article presents a regularization method based on optimization form of the Gauss principle of least constraint and the Udwadia and Kalaba (UK) method, which expresses the solution of the dynamic motion in an explicit expression. Compared with Lagrange method, the proposed method does not need to solve the Lagrange multipliers any more for the complex multibody mechanical systems. In the end, a numerical example is given to illustrate that the proposed method not only has the ability to handle the problems with redundant constraints, but also has higher accuracy and keep a stable constraint violation level than the traditional methods.

Non-Iterative Multi-Area Coordinated Dispatch via Condensed System Representation

The coordinated multi-area economic dispatch enables optimal resource utilization in a larger spatial range. To overcome drawbacks such as heavy communication burden, convergence failure, and weak scalability of iterative coordination methods, this paper proposes a fully non-iterative multi-area coordination framework. The proposed framework is based on a novel system reduction technique termed as the condensed system representation (CSR). The CSR makes external equivalence of the dispatch problem of each area by exploiting power system operating characteristics, i.e., only a small number of generators may become marginal units and only a minority of security constraints may be active. An algorithm based on the optimality condition analysis is developed to identify the CSR by fixing non-marginal units to their output bounds, eliminating redundant security constraints, and making Norton equivalence of the internal network. With CSRs submitted by local areas, the multi-area system can be optimized without iterative information exchange. Case studies based on a 3-area 9-bus system verify the effectiveness of the CSR-based coordination framework. Larger-scale test systems are constructed to validate the computational efficiency, robustness, and scalability of the proposed method.

Utilizing the redundant constraints for the uplift payment elimination

A power market with non-convexities may not have an equilibrium price for power that provides economic stability of the centralized dispatch outcome. In this case, the market players are entitled to receive the uplift payments that compensate the economic profit lost when following the centralized dispatch. We consider a special class of the (possibly non-linear) redundant constraints that are redundant not only on the feasible set of the centralized dispatch optimization problem (and, therefore, do not change the centralized dispatch outcome) but also on the larger set obtained when the power balance constraint is relaxed. We show that the Lagrangian relaxation of these redundant constraints may reduce the uplift payments without changing the duality gap. For any given market price (or a pricing algorithm that sets the producer revenue as a function of its output volume) in a uninode multi-period power market with fixed load, we explicitly construct a family of the redundant constraints that do not change the maximum profit of the producer and result in zero uplift payment. We show that the introduction and subsequent Lagrangian relaxation of just one redundant constraint in the centralized dispatch problem suffice to eliminate the uplift payments for all the producers. In the case of the convex hull pricing method, the introduction of these redundant constraints affects neither the duality gap nor the market price for power. The results can be straightforwardly generalized to a power market with the price-sensitive load.

Least squares KNN-based weighted multiclass twin SVM

K-nearest neighbor (KNN) based weighted multi-class twin support vector machines (KWMTSVM) is a novel multi-class classification method. In this paper, we propose a novel least squares version of KWMTSVM called LS-KWMTSVM by replacing the inequality constraints with equality constraints and minimized the slack variables using squares of 2-norm instead of conventional 1-norm. This simple modification leads to a very fast algorithm with much better results. The modified primal problems in the proposed LS-KWMTSVM solves only two systems of linear equations whereas two quadratic programming problems (QPPs) need to solve in KWMTSVM. The proposed LS-KWMTSVM, same as KWMTSVM, employed the weight matrix in the objective function to exploit the local information of the training samples. To exploit the inter class information, we use weight vectors in the constraints of the proposed LS-KWMTSVM. If any component of vectors is zero then the corresponding constraint is redundant and thus we can avoid it. Elimination of redundant constraints and solving a system of linear equations instead of QPPs makes the proposed LS-KWMTSVM more robust and faster than KWMTSVM. The proposed LS-KWMTSVM, commensurate as the KWMTSVM, all the training data points into a “1-versus-1-versus-rest” structure, and thus our LS-KWMTSVM generate ternary output {-1,0,+1} which helps to deal with imbalance datasets. Numerical experiments on several UCI and KEEL imbalance datasets(with high imbalance ratio) clearly indicate that the proposed LS-KWMTSVM has better classification accuracy compared with other baseline methods but with remarkably less computational time.

Decentralized DLMPs with synergetic resource optimization and convergence acceleration

Dynamic pricing has been proposed for scheduling and controlling distributed energy resources (DERs) to mitigate operational problems in distribution networks. In this paper, we employ distribution locational marginal prices (DLMPs) to optimally schedule DERs, considering both line and voltage constraints. DLMPs are calculated in an iterative, decentralized manner to respect user privacy, using dual decomposition and the subgradient method. The inclusion of generating DERs in the problem formulation, and the nature of the subgradient method, may significantly increase the amount of required iterations. Recognizing the importance of convergence speed in such a mechanism, we act upon this by proposing a novel concept to identify and remove redundant constraints of the problem. The method is based on network topology observations in radial distribution networks. Redundant constraints are mapped to the respective Lagrange multipliers and are set to zero, thereby significantly increasing convergence speed. We validate our method in a 33 and a 136 bus system, showing the synergies of the co-optimization of generating and consuming DERs, and reducing the number of iterations at least threefold.