Large Number Of 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.

Varying Condition SCOPF Optimization Based on Deep Learning and Knowledge Graph

Security-constrained optimal power flow (SCOPF) is a vital task for independent system operators (ISO) in daily scheduling. However, the large number of inequality constraints bring us big challenges to solve large-scale SCOPF in real time. This paper proposes a fast solution method for SCOPF by predicting active constraints based on machine learning approaches. Namely, deep neural networks (DNNs) are employed to predict active security constraints based on historical data, thus accelerating the SCOPF calculation. Active margin functions are proposed to quantify how likely these security constraints will be active, thus improving our prediction accuracy. Knowledge graph is adopted to record system working conditions, pertinent learning results and their relationship, thus improving the transferability of the learning model under varying operation conditions. Simulations have been done on IEEE 30-bus, 118-bus and 300-bus systems to demonstrate the effectiveness of the proposed DNN approach. The influence of artificial parameters and the effectiveness of the knowledge graph are also illustrated.

IJIREEICE - Electrical, Electronics, Instrumentation and Control

The operation of generating units to generate energy at reduced cost to reliably serve consumers, recognizing any operational restrictions of generation and transmission facilities is called as Economic Dispatch(ED).In the present paper, an evolutionary modified Deterministic Particle Swarm Optimization (DPSO) is applied for Environmental and Economic Power Dispatch (EED) problem to find fast and efficient solutions for IEEE 14 and 30 bus systems. In such a way, proposed algorithms reduce the total expenditure task of the generating units and improve the computational efficiency. The feasibility of the proposed DPSO based algorithm is demonstrated for two power system test cases consisting of IEEE 14 bus 3-generator system and 30 bus 6-generator system. The results obtained are compared to previous PSO, GA. In practical, power systems which are able to supply a bounded range of electrical load demand, optimizing the operation expenses of the generation units is very significant from an economic point of view. It can be done by proper scheduling of each generating units which is commonly called as Economic dispatch. The conservative technique includes lambda iteration method, base point and participation factor method, gradient method, etc. However, these conventional dispatch algorithms necessitate the incremental cost curves to be monotonically rising or piece-wise linear and are highly sensitive in selecting the initial point and frequently converge to local optimal solution or diverge altogether. Hence, Economic Dispatch (ED) techniques are used to determine a condition with the lowest generation costs. The load demand, transmission power losses and generation cost coefficients are the factors must be taken into account for any ED technique. Newton based algorithms have a problem in handling a large number of inequality constraints. Linear programming methods are fast and trustworthy, but the main drawback is linked with the piecewise linear cost approximation. Non linear encoding technique has a problem of convergence and algorithm complexity.

Power System Reactive Power Optimization Based on Fuzzy Formulation and Interior Point Filter Algorithm

Considering the soft constraint characteristics of voltage constraints, the Interior-Point Filter Algorithm is applied to solve the formulation of fuzzy model for the power system reactive power optimization with a large number of equality and inequality constraints. Based on the primal-dual interior-point algorithm, the algorithm maintains an updating “filter” at each iteration in order to decide whether to admit correction of iteration point which can avoid effectively oscillation due to the conflict between the decrease of objective function and the satisfaction of constraints and ensure the global convergence. Moreover, the “filter” improves computational efficiency because it filters the unnecessary iteration points. The calculation results of a practical power system indicate that the algorithm can effectively deal with the large number of inequality constraints of the fuzzy model of reactive power optimization and satisfy the requirement of online calculation which realizes to decrease the network loss and maintain specified margins of voltage.

Particle Swarm Optimization and Bacterial Foraging Optimization Techniques for Optimal Current Harmonic Mitigation by Employing Active Power Filter

Conventional mathematical modeling-based approaches are incompetent to solve the electrical power quality problems, as the power system network represents highly nonlinear, nonstationary, complex system that involves large number of inequality constraints. In order to overcome the various difficulties encountered in power system such as harmonic current, unbalanced source current, reactive power burden, active power filter (APF) emerged as a potential solution. This paper proposes the implementation of particle swarm optimization (PSO) and bacterial foraging optimization (BFO) algorithms which are intended for optimal harmonic compensation by minimizing the undesirable losses occurring inside the APF itself. The efficiency and effectiveness of the implementation of two approaches are compared for two different conditions of supply. The total harmonic distortion (THD) in the source current which is a measure of APF performance is reduced drastically to nearly 1% by employing BFO. The results demonstrate that BFO outperforms the conventional and PSO-based approaches by ensuring excellent functionality of APF and quick prevail over harmonics in the source current even under unbalanced supply.

On the Use of Outer Approximations as an External Active Set Strategy

Outer approximations are a well known technique for solving semiinfinite optimization problems. We show that a straightforward adaptation of this technique results in a new, external, active-set strategy that can easily be added to existing software packages for solving nonlinear programming problems with a large number of inequality constraints. Our external active-set strategy is very easy to implement, and, as our numerical results show, it is particularly effective when applied to discretized semiinfinite optimization or state-constrained optimal control problems. Its effects can be spectacular, with reductions in computing time that become progressively more pronounced as the number of inequalities is increased.

An External Active-Set Strategy for Solving Optimal Control Problems

We present a new, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">external</i> , active constraints set strategy for solving nonlinear programming problems with a large number of inequality constraints that arise in the process of discretizing continuous-time optimal control problems with state-space constraints. This strategy constructs a sequence of inequality constrained nonlinear programming problems, containing a progressively larger subset of the constraints in the original problem, and submits them to a nonlinear programming solver for a fixed number of iterations. We prove that this scheme computes a solution of the original problem and show by means of numerical experiments that this strategy results in reductions in computing time ranging from a factor of 6 to a factor of over 100.

An Accelerator for Packages Solving Discrete-Time Optimal Control Problems

We present an accelerator scheme for use with existing packages that solve nonlinear programming problems with a large number of inequality constraints that arise in the process of discretizing continuous-time optimal control problems with state-space constraints. This scheme is based on the concept of outer approximations used in semi-infinite programming and acts as an external, active constraints set strategy.Our scheme constructs a finite sequence of inequality constrained nonlinear programming problems, containing a progressively larger subset of the constraints in the original problem, and submits these problems to a nonlinear programming solver for a fixed number of iterations. We prove that this scheme computes a solution of the original problem and show, by means of numerical experiments, that it results in reductions in computing time ranging from a factor of 6 to a factor of over 400.

A solution method of unit commitment by artificial neural networks

The authors explore the possibility of applying the Hopfield neural network to combinatorial optimization problems in power systems, in particular to unit commitment. A large number of inequality constraints included in unit commitment can be handled by dedicated neural networks. As an exact mapping of the problem onto the neural network is impossible with the state of the art, a two-step solution method was developed. First, generators to be stored up at each period are determined by the network, and then their outputs are adjusted by a conventional algorithm. The proposed neural network could solve a large-scale unit commitment problem with 30 generators over 24 periods, and results obtained were very encouraging. >

AN OPTIMIZATION METHOD USING SELECTION TECHNIQUES OF ACTIVE CONSTRAINT SURFACES

There are a large number of inequality constraints in a structural optimum design problem, however, an optimum design point is usually located on active constraint surfaces that are equivalent to equality constraint. If we select such equality constraints among the inequality constraints, the inequality constrained problem can be transformed into an equality constrained problem. An optimum point of the new problem that is subjected to the selected equality constraints is obtained by using the Lagrange multiplier technique. In this paper an optimization method in which selection techniques are used is proposed, and a simple example on optimum solutions of composite plate girders is shown.

A sequential linear optimization approach for controller design

A linear optimization approach with a simple, real arithmetic algorithm is presented for reliable controller design and vibration suppression of flexible structures. Using first-order sensitivity of the system eigenvalues with respect to the design parameters in conjunction with a continuation procedure, the method converts a nonlinear optimization problem into a maximization problem with linear inequality constraints. The method of linear programming is then applied to solve the converted linear optimization problem. The general efficiency of the linear programming approach allows the method to handle structural optimization problems with a large number of inequality constraints on the design vector. The method is demonstrated using a truss beam, finite element model for the optimal sizing and placement of active/passive structural members for damping augmentation. Results using both the sequential linear optimization approach and nonlinear optimization are presented and compared. The insensitivity to the initial conditions of the linear optimization approach is also demonstrated.

An efficient feasible directions algorithm for design synthesis

A nonlinear optimization algorithm is described that combines the best features of the method of feasible directions and the generalized reduced gradient method. The algorithm uses the direction-finding subproblem from the method of feasible directions to find a search direction that is equivalent to that of the generalized reduced gradient method, but without the need to add a large number of slack variables associated with inequality constraints. This leads to a core-efficient algorithm for the solution of optimization problems with a large number of inequality constraints. Also, during the one-dimensional search, it is not necessary to separate the design space into dependent and independent variables using the present method. The concept of infrequent gradient calculations is introduced as a means of gaining further optimization efficiency. Finally, it is shown that, using the basic direction-finding algorithm contained in this method, the sensitivity of the optimum design with respect to problem parameters can be obtained without the need for second derivatives or Lagrange multipliers. The optimization algorithm and sensitivity analysis is demonstrated by numerical example.