Discrete Constrained Optimization Problems Research Articles

Quantum entanglement inspired hard constraint handling for operations engineering optimization with an application to airport shift planning

Constraint handling, especially hard constraint handling, is a critical issue for discrete constrained optimization problem solving in operations engineering. The performance of conventional constraint handling techniques, such as penalty functions and repair operators, are susceptible to problem complexity and scale. The recent dawning of quantum computation has been recognized as a very promising direction in the paradigm of optimization. This paper presents a quantum entanglement inspired hard constraint handling approach, to efficiently solve discrete constrained optimization problems. We propose a quantum entanglement inspired representation method of hard constraints, which uses a series of qubits and quantum circuit to coherently represent the hard constraints in the posterior quantum states. A quantum entanglement inspired genetic algorithm (QEI-GA) is proposed by embedding the quantum entanglement inspired representation into classical genetic algorithm optimization strategy. The proposed method is applied to solve a real-world airport workforce shift planning problem to validate the algorithm’s effectiveness and robustness. The results of the case study show that the proposed method can consistently reach better solutions for the discrete constrained optimization problem, even with a much smaller population size, compared to the conventional genetic algorithms with penalty function based and multi-objective concept-based strategies.

An augmented Lagrangian proximal alternating method for sparse discrete optimization problems

In this paper, we propose an augmented Lagrangian proximal alternating (ALPA) method for solving two classes of large-scale sparse discrete constrained optimization problems. Specifically, the ALPA method generates a sequence of augmented Lagrangian (AL) subproblems in the out iterations and utilizes a proximal alternating linearized minimization method and sparse projection techniques to solve these AL subproblems. And we study the first-order optimality conditions for these two classes of problems. Under some suitable assumptions, we show that any accumulation point of the sequence generated by the ALPA method satisfies the necessary first-order optimality conditions of these problems or is a local minimizer of these problems. The computational results with practical problems demonstrate that our method can find the suboptimal solutions of the problems efficiently and is competitive with some other local solution methods.

Axis-Aligned Height-Field Block Decomposition of 3D Shapes

We propose a novel algorithm for decomposing general three-dimensional geometries into a small set of overlap-free height-field blocks , volumes enclosed by a flat base and a height-field surface defined with respect to this base. This decomposition is useful for fabrication methodologies such as 3-axis CNC milling, where a single milling pass can only carve a single height-field surface defined with respect to the machine tray but can also benefit other fabrication settings. Computing our desired decomposition requires solving a highly constrained discrete optimization problem, variants of which are known to be NP-hard. We effectively compute a high-quality decomposition by using a two-step process that leverages the unique characteristics of our setup. Specifically, we notice that if the height-field directions are constrained to the major axes, then we can always produce a valid decomposition starting from a suitable surface segmentation. Our method first produces a compact set of large, possibly overlapping, height-field blocks that jointly cover the model surface by recasting this discrete constrained optimization problem as an unconstrained optimization of a continuous function, which allows for an efficient solution. We then cast the computation of an overlap-free, final decomposition as an ordering problem on a graph and solve it via a combination of cycle elimination and topological sorting. The combined algorithm produces a compact set of height-field blocks that jointly describe the input model within a user given tolerance. We demonstrate our method on a range of inputs and showcase a number of real life models manufactured using our technique.

Cohort intelligence algorithm for discrete and mixed variable engineering problems

In this study, the performance of an emerging socio inspired metaheuristic optimization technique referred to as Cohort Intelligence (CI) algorithm is evaluated on discrete and mixed variable nonlinear constrained optimization problems. The investigated problems are mainly adopted from discrete structural optimization and mixed variable mechanical engineering design domains. For handling the discrete solution variables, a round off integer sampling approach is proposed. Furthermore, in order to deal with the nonlinear constraints, a penalty function method is incorporated. The obtained results are promising and computationally more efficient when compared to the other existing optimization techniques including a Multi Random Start Local Search algorithm. The associated advantages and disadvantages of CI algorithm are also discussed evaluating the effect of its two parameters namely the number of candidates, and sampling space reduction factor.

A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation

AbstractWe study the TV-L1 image approximation model from primal and dual perspective, based on a proposed equivalent convex formulations. More specifically, we apply a convex TV-L1 based approach to globally solve the discrete constrained optimization problem of image approximation, where the unknown image function u(x) ∈ {f1,…,fn}, ∀x ∈ Ω. We show that the TV-L1 formulation does provide an exact convex relaxation model to the non-convex optimization problem considered. This result greatly extends recent studies of Chan et al., from the simplest binary constrained case to the general gray-value constrained case, through the proposed rounding scheme. In addition, we construct a fast multiplier-based algorithm based on the proposed primal-dual model, which properly avoids variability of the concerning TV-L1 energy function. Numerical experiments validate the theoretical results and show that the proposed algorithm is reliable and effective.

Multilocus consensus genetic maps (MCGM): Formulation, algorithms, and results

In process of creating genetic maps different labs/research groups obtain overlapping parts of the map. Merging these parts into one integrative map is based on looking for maximum shared marker orders among the maps. Really, not all shared markers of such maps have consensus order that obstructs building of the integrative maps. In this paper we propose a new approach to build verified multilocus consensus genetic maps in which shared markers always are integrated in stable consensus order. The approach is based on combined analysis of initial mapping data rather than manipulating with previously constructed maps. We show that more effective and reliable solutions may be obtained based on “synchronized ordering” facilitated by cycles of “re-sampling → ordering → removing unstable markers”. The proposed formulation of consensus genetic mapping can be considered as a version of traveling salesperson problem (TSP) that we refer to as synchronized-TSP. From the viewpoint of optimization, synchronized-TSP belongs to discrete constrained optimization problems. Earlier we developed new powerful and fast guided evolution strategy algorithms for some types of discrete constrained optimization. These algorithms were used here as a basis for solving more challenging problems of consensual marker ordering.

Penalty Formulations and Trap-Avoidance Strategies for Solving Hard Satisfiability Problems

In this paper we study the solution of SAT problems formulated as discrete decision and discrete constrained optimization problems. Constrained formulations are better than traditional unconstrained formulations because violated constraints may provide additional forces to lead a search towards a satisfiable assignment. We summarize the theory of extended saddle points in penalty formulations for solving discrete constrained optimization problems and the associated discrete penalty method (DPM). We then examine various formulations of the objective function, choices of neighborhood in DPM, strategies for updating penalties, and heuristics for avoiding traps. Experimental evaluations on hard benchmark instances pinpoint that traps contribute significantly to the inefficiency of DPM and force a trajectory to repeatedly visit the same set of or nearby points in the original variable space. To address this issue, we propose and study two trap-avoidance strategies. The first strategy adds extra penalties on unsatisfied clauses inside a trap, leading to very large penalties for unsatisfied clauses that are trapped more often and making these clauses more likely to be satisfied in the future. The second strategy stores information on points visited before, whether inside traps or not, and avoids visiting points that are close to points visited before. It can be implemented by modifying the penalty function in such a way that, if a trajectory gets close to points visited before, an extra penalty will take effect and force the trajectory to a new region. It specializes to the first strategy because traps are special cases of points visited before. Finally, we show experimental results on evaluating benchmarks in the DIMACS and SATLIB archives and compare our results with existing results on GSAT, WalkSAT, LSDL, and Grasp. The results demonstrate that DPM with trap avoidance is robust as well as effective for solving hard SAT problems.

HYBRID EVOLUTIONARY AND ANNEALING ALGORITHMS FOR NONLINEAR DISCRETE CONSTRAINED OPTIMIZATION

This paper presents a procedural framework that unifies various mechanisms to look for discrete-neighborhood saddle points in solving discrete constrained optimization problems (DCOPs). Our approach is based on the necessary and sufficient condition on local optimality in discrete space, which shows the one-to-one correspondence between the discrete-space constrained local minima of a problem and the saddle points of the corresponding Lagrangian function. To look for such saddle points, we study various mechanisms for performing ascents of the Lagrangian function in the original-variable subspace and descents in the Lagrange-multiplier subspace. Our results show that CSAEA, a combined constrained simulated annealing and evolutionary algorithm, performs well when using mutations and crossovers to generate trial points and accepting them based on the Metropolis probability. We apply iterative deepening to determine the optimal number of generations in CSAEA and show that its performance is robust with respect to changes in population size. To test the performance of the procedures developed, we apply them to solve some continuous and mixed-integer nonlinear programming (NLP) benchmarks and show that they obtain better results than those of existing algorithms.

A Design Method of an Automotive Wheel-Bearing Unit With Discrete Design Variables Using Genetic Algorithms

In order to improve the efficiency of the design process and the quality of the resulting design, this study proposes a design method for determining design variables of an automotive wheel-bearing unit of double-row angular-contact ball bearing type by using a genetic algorithm. The desired performance of the wheel-bearing unit is to maximize system life while satisfying geometrical and operational constraints without enlarging mounting space. The design variables selected are number of balls, initial contact angle, standard ball diameter, pitch circle diameter, preload, distance between ball centers, and wheel offset, which must be selected at the preliminary design stage. The use of gradient-based optimization methods for the design of the unit is restricted because this design problem is characterized by the presence of discrete design variables such as the number of balls and standard ball diameter. Therefore, the design problem of rolling element bearings is a constrained discrete optimization problem. A genetic algorithm using real coding and dynamic mutation rate is used to efficiently find the optimum discrete design values. To effectively deal with the design constraints, a ranking method is suggested for constructing a fitness function in the genetic algorithm. A computer program is developed and applied to the design of a real wheel-bearing unit model to evaluate the proposed design method. Optimum design results demonstrate the effectiveness of the design method suggested in this study by showing that the system life of an optimally designed wheel-bearing unit is enhanced in comparison with that of the current design without any constraint violations. It is believed that the proposed methodology can be applied to other rolling element bearing design applications.

Design Optimization of Deep Groove Ball Bearing with Discrete Variables for High-Load Capacity

A design method for maximizing fatigue life of the deep groove ball bearing without enlarging mounting space is proposed by using a genetic algorithm. The use of gradient-based optimization methods for the design of the bearing is restricted because this design problem is characterized by the presence of discrete design variables such as the number of balls and standard ball diameter. Therefore, the design problem of rolling element bearings is a constrained discrete optimization problem. A genetic algorithm using real coding is used to efficiently find the optimum discrete design values. To effectively deal with the design constraints, a ranking method is suggested for constructing a fitness function in the genetic algorithm. Constrains for manufacturing are applied in optimization scheme. Results obtained for several 63 series deep groove ball bearings demonstrated the effectiveness of the proposed design methodology by showing that the average basic dynamic capacities of optimally designed bearings increased about 9-34% compared with the standard ones.

Discrete Lagrangian methods for optimizing the design of multiplierless QMF banks

In this paper, we present a new discrete Lagrangian method for designing multiplierless quadrature mirror filter banks. The filter coefficients in these filter banks are in powers-of-two, where numbers are represented as sums or differences of powers of two (also called canonical signed digit representation), and multiplications are carried out as additions, subtractions, and shifts. We formulate the design problem as a nonlinear discrete constrained optimization problem, using reconstruction error as the objective, and stopband and passband energies, stopband and passband ripples, and transition bandwidth as constraints. Using the performance of the best existing designs as constraints, we search for designs that improve over the best existing designs with respect to all the performance metrics. We propose a new discrete Lagrangian method for finding good designs and study methods to improve the convergence speed of Lagrangian methods without affecting their solution quality. This is done by adjusting dynamically the relative weights between the objective and the Lagrangian part. We show that our method can find designs that improve over Johnston's benchmark designs using a maximum of three to six ONE bits in each filter coefficient instead of using floating-point representations. Our approach is general and is applicable to the design of other types of multiplierless filter banks.

Decentralized computation procurement and computational robustness in a smart market

Several `smart market' mechanisms have recently appeared in the literature. These mechanisms combine a computer network that collects bids from agents with a central computer that selects a schedule of bids to fill based upon maximization of revenue or trading surplus. Potential problems exist when this optimization involves combinatorial difficulty sufficient to overwhelm the central computer. This paper explores the use of a computation procuring clock auction to induce human agents to approximate the solutions to discrete constrained optimization problems. Economic and computational properties of the auction are studied through a series of laboratory experiments. The experiments are designed around a potential application of the auction as a secondary institution that approximates the solution to difficult computational problems that occur within the primary `smart market', and show that the auction is effective and robust in eliciting and processing suggestions for improved schedules.

DS-CDMA chip waveform design for minimal interference under bandwidth, phase, and envelope constraints

This paper investigates the effect of chip waveform shaping on the error performance, bandwidth confinement, phase continuity, and envelope uniformity in direct-sequence code-division multiple-access communication systems employing offset quadrature modulation formats. An optimal design methodology is developed for the problem of minimizing the multiple-access interference power under various desirable signal constraints, including limited 99% and 99.9% power bandwidth occupancies, continuous signal phase, and near-constant envelope. The methodology is based on the use of prolate spheroidal wave functions to obtain a reduced-dimension discrete constrained optimization problem formulation. Numerous design examples are discussed to compare the performance achieved by the optimally-designed chip waveforms with other conventional schemes, such as offset quadrature phase-shift keying, minimum-shift keying (MSK), sinusoidal frequency-shift keying (SFSK), and time-domain raised-cosine pulses. In general, it is found that while the optimized chip pulses achieved substantial gains when no envelope constraints were imposed, these gains vanish when a low envelope fluctuation constraint was introduced. In particular, it is also shown that MSK is quasi-optimal with regard to the 99% bandwidth measure, while the raised-cosine pulse is equally good with both the 99% and 99.9% measures, but at the expense of some envelope variation. On the other hand, SFSK is quasi-optimal with regard to the 99.9% bandwidth occupancy, among the class of constant-to-low envelope variation pulses.

Enhanced genetic operators for the resolution of discrete constrained optimization problems

The behavior of the Crossover and Mutation Operators on candidate solutions to any discrete and constrained optimization problem is investigated in some detail, affording new understanding in the construction of Evolutionary and Genetic Algorithms for the effective resolution of problems of this kind. The important case in which some or all of the constraints are linear exemplifies the potential usefulness of these insights, by guiding the design and application of operators guaranteed to preserve feasibility. The computational challenge posed by numerous optimization problems has lead to the exploration of algorithms based on adaption observed in biological systems, as an alternative means of illation. Ideally suited to difficult conditions, in accordance with their inspiration, these methods have been successfully exploited in many problems that have defied adequate resolution by more traditional means. Despite their undeniable attraction, the applicability of Genetic Algorithms has been limited by the lack of general techniques to manage systems of constraints, a feature of many optimization tasks, and this is compounded when some proportion of the decision variables are discrete. In response, the fundamental behavior of the genetic operators with respect to any system of constraints has been investigated; this discussion summarizes and extends the results of earlier work. [ Math. Comp. Modelling, 1996, 23(5), 87–111]. By investigating theoretically the behavior of the crossover operator on viable solutions of an entirely general system of constraints, the limitations of restricting attention to solely to feasible points become apparent. A relaxation of the crossover operator is also proposed to avoid calamitous performance degradation otherwise experienced in solving highly constrained problems, and its effective utilization suggests a modification in the overall algorithm in which the population is permitted to transiently increase in size.