PN to CSP Methodology: Improved Bounds

Optimizing the operation of a multireservoir system is challenging due to the high dimension of the decision variables that lead to a large and complex search space. A spectral optimization model (SOM), which transforms the decision variables from time domain to frequency domain, is proposed to reduce the dimensionality. The SOM couples a spectral dimensionality‐reduction method called Karhunen‐Loeve (KL) expansion within the routine of Nondominated Sorting Genetic Algorithm (NSGA‐II). The KL expansion is used to represent the decision variables as a series of terms that are deterministic orthogonal functions with undetermined coefficients. The KL expansion can be truncated into fewer significant terms, and consequently, fewer coefficients by a predetermined number. During optimization, operators of the NSGA‐II (e.g., crossover) are conducted only on the coefficients of the KL expansion rather than the large number of decision variables, significantly reducing the search space. The SOM is applied to the short‐term operation of a 10‐reservoir system in the Columbia River of the United States. Two scenarios are considered herein, the first with 140 decision variables and the second with 3360 decision variables. The hypervolume index is used to evaluate the optimization performance in terms of convergence and diversity. The evaluation of optimization performance is conducted for both conventional optimization model (i.e., NSGA‐II without KL) and the SOM with different number of KL terms. The results show that the number of decision variables can be greatly reduced in the SOM to achieve a similar or better performance compared to the conventional optimization model. For the scenario with 140 decision variables, the optimal performance of the SOM model is found with six KL terms. For the scenario with 3360 decision variables, the optimal performance of the SOM model is obtained with 11 KL terms.

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

This paper presents a VLSI design for a competitive neural network model, known as GENET (Wang and Tsang 1991), for solving Constraint Satisfaction Problems (CSP). The CSP is a mathematical abstraction of the problems in many AI application domains. In essence, a CSP can be defined as a triple (Z, D, C), where Z is a finite set of variables, D is a mapping from every variable to a domain, which is a finite set of arbitrary objects, and C is a set of constraints. Each constraint in C restricts the values that can be simultaneously assigned to a number of variables in Z. If the constraints in C involve up to but no more than n variables it is called an n-ary CSP. The task is to assign one value per variable satisfying all the constraints in C (Mackworth 1977). In addition, associated with the variable assignments might be costs and utilities. This turns CSPs into optimization problems, demanding CSP solvers to find a set of variable assignments that would produce a maximum total utility at a minimal cost. Furthermore, some CSPs might be over-constrained, i.e. not all the constraints in C can be satisfied simultaneously. In this case, the set of assignments to a maximum number of variables without violating any constraints in C, or the set of assignments to all the variables which violates a minimal number of constraints might be sought for.

Many real-world multi-objective optimization problems have hundreds or even thousands of decision variables, which contrast with the current practice of multi-objective metaheuristics whose performance is typically assessed using benchmark problems with a relatively low number of decision variables (normally, no more than 30). In this paper, we propose a cooperative coevolution framework that is capable of optimizing large scale (in decision variable space) multi-objective optimization problems. We adopt a benchmark that is scalable in the number of decision variables (the ZDT test suite) and compare our proposed algorithm with respect to two state-of-the-art multi-objective evolutionary algorithms (GDE3 and NSGA-II) when using a large number of decision variables (from 200 up to 5000). The results clearly indicate that our proposed approach is effective as well as efficient for solving large scale multi-objective optimization problems.

Multi-objective evolutionary algorithms (MOEAs) of the state of the art are created with the only purpose of dealing with the number of objective functions in a multi-objective optimization problem (MOP) and treat the decision variables of a MOP as a whole. However, when dealing with MOPs with a large number of decision variables (more than 100) their efficacy decreases as the number of decision variables of the MOP increases. On the other hand, problem decomposition, in terms of decision variables, has been found to be extremely efficient and effective for solving large scale optimization problems. Nevertheless, most of the currently available approaches for large scale optimization rely on models based on cooperative coevolution or linkage learning methods that use multiple subpopulations or preliminary analysis, respectively, which is computationally expensive (in terms of function evaluations) when used within MOEAs. In this work, we study the effect of what we call operational decomposition, which is a novel framework based on coevolutionary concepts to apply MOEAs’s crossover operator without adding any extra cost. We investigate the improvements that NSGA-III can achieve when combined with our proposed coevolutionary operators. This new scheme is capable of improving efficiency of a MOEA when dealing with large scale MOPs having from 200 up to 1200 decision variables.

Large-scale high-dimensional many-objective optimization problems (LaMaOPs) are prevalent in fields such as autonomous driving, cloud resource scheduling, and smart grids. LaMaOPs involve a large number of decision variables and multiple conflicting objectives that need to be optimized simultaneously. The challenges posed by the curse of dimensionality due to the vast number of decision variables, and the conflict between convergence and diversity caused by the numerous objective variables, make traditional optimization methods inadequate. To address these issues, this paper proposes a two-population cooperative evolutionary algorithm based on large-scale decision variable analysis (DVA-TPCEA). This algorithm integrates quantitative analysis methods for decision variables to deeply examine their impact on each objective and introduces a contribution-based objective detection method. Additionally, a dual-population cooperative evolution mechanism is employed, with targeted optimization strategies designed for convergence and diversity populations, achieving synergistic and complementary optimization between the two populations. To validate the algorithm’s effectiveness in practical applications, a large-scale container resource scheduling strategy based on the DVA-TPCEA algorithm is also proposed. The experimental results indicate that the proposed algorithm demonstrates significant advantages in both general datasets DTLZ, WFG, and LSMOP, and practical models.

Optimal classification trees with leaf-branch and binary constraints

Gradient-free methods applied to optimisation of advanced ultra-supercritical power plant

This article studies a joint stocking and product offer problem. We have access to a number of products to satisfy the demand over a finite selling horizon. Given that customers choose among the set of offered products according to the multinomial logit model, we need to decide which sets of products to offer over the selling horizon and how many units of each product to stock so as to maximize the expected profit. We formulate the problem as a nonlinear program, where the decision variables correspond to the stocking quantity for each product and the duration of time that each set of products is offered. This nonlinear program is intractable due to its large number of decision variables and its nonseparable and nonconcave objective function. We use the structure of the multinomial logit model to formulate an equivalent nonlinear program, where the number of decision variables is manageable and the objective function is separable. Exploiting separability, we solve the equivalent nonlinear program through a dynamic program with a two dimensional and continuous state variable. As the solution of the dynamic program requires discretizing the state variable, we study other approximate solution methods. Our equivalent nonlinear program and approximate solution methods yield insights for good offer sets.

The widely used support vector machine (SVM) method has shown to yield very good results in supervised classification problems. Other methods such as classification trees have become more popular among practitioners than SVM thanks to their interpretability, which is an important issue in data mining.In this work, we propose an SVM-based method that automatically detects the most important predictor variables and the role they play in the classifier. In particular, the proposed method is able to detect those values and intervals that are critical for the classification. The method involves the optimization of a linear programming problem in the spirit of the Lasso method with a large number of decision variables. The numerical experience reported shows that a rather direct use of the standard column generation strategy leads to a classification method that, in terms of classification ability, is competitive against the standard linear SVM and classification trees. Moreover, the proposed method is robust; i.e., it is stable in the presence of outliers and invariant to change of scale or measurement units of the predictor variables.When the complexity of the classifier is an important issue, a wrapper feature selection method is applied, yielding simpler but still competitive classifiers.

To support national carbon neutrality goals, China's grid is rapidly decarbonizing. Confronting power volatility and renewable curtailment induced by large-scale wind/PV penetration, this paper introduces an optimal scheduling framework for multi-energy complementary systems integrating thermal, storage, and renewables. Considering the complexity of the multi-energy structure, which involves a large number of decision variables and operational constraints, the model is linearized, and a hierarchical optimization framework is employed. The upper-level model minimizes thermal, storage, and renewable operating costs, using storage for peak-shaving to reduce load fluctuation and enable renewable integration. The lower level minimizes the thermal and storage costs, and curtailment penalties to boost renewable utilization and operational economy. A decomposition-and-coordination solution methodology is adopted to achieve iterative coordination between the two model layers. Finally, the proposed strategy is validated through simulation studies based on an improved IEEE 39-bus test system under multiple scenarios. Simulation results confirm that the proposed methodology substantially improves renewable energy accommodation capacity and overall system economic efficiency, validating the model's effectiveness and practical applicability.

Molecular sequence alignment is one of the most essential tools of the molecular biology. It permits to track changes and similarities between molecular sequences. In this paper the molecular sequence alignment problem is formulated suitable for an evolutionary algorithm (EA), and two problem instances are solved using generalized differential evolution 3 (GDE3), which is a general purpose EA. Regardless of relatively large number of decision variables, the instances were solvable and results were comparable to those by sequence alignment solvers in comparison

The particle swarm optimization (PSO) algorithm is a random search algorithm that simulates biological activities in nature and swarm intelligence. Traditional PSO algorithms still suffer from their inherent problems such as long computing time and premature convergence when dealing with a large number of decision variables. How to reduce the computing time while ensuring its accuracy is one of the main research issues in the literature. This paper proposes a PSO algorithm with matrix-based fuzzy adaptation and group learning. Firstly, to reduce the computing time, we propose a strategy to balance the global search ability and the local search ability by controlling the number of iterations adaptively. Second, to accelerate the convergence speed of the algorithm, we further propose a group learning mechanism. Along with the above two considerations, the proposed algorithm is able to achieve a promising balance between exploration and exploitation and, consequently, improves the performance. Through extensive comparison experiments on six classic functions, we show that our proposed PSO algorithm can achieve competitive or even better performance than several state-of-the-art methods.

The low recovery factor associated with carbonate reservoir and the availability of affordable gas has drawn new attention to a Water Alternative Gas (WAG) strategy to enhance hydrocarbon recovery for reservoir at early stage of production. The use of WAG allows to improve sweep efficiency limiting fingering and hydrocarbon trapping at macroscopic (Water injection - WI) and microscopic (pore) (Gas Injection - GI) level.The purpose of this work is twofold: present a new field development plan optimization formulation generalized for continuous and/or WAG injection and its suggested application to a real reservoir test case. Optimum field development plans for different strategies are presented and the advantages of a WAG strategy as opposite to a continuous water injection are further discussed. The proposed optimization problem covers well placement, control, schedule and gas lift under uncertainty. This problem is formulated as Mixed Integer Nonlinear Programing (MINLP) with non-linear constrains in order to take into account operational restrictions.A carbonate deep water reservoir at the early stages of its exploitation has been used to evaluate this new workflow. An in-house Particle Swarm Optimization (PSO) algorithm has been used to solve this nonlinear optimization problem. The use of a real field as test bench poses additional strength on the robustness of the formulation in presence of a large number of decision variables and constrains. Special care has been taken in the field development plan selection, taking into account the optimal solution as well as additional objectives and sentiments added by the decision makers which increase the realism of the final solution.The optima selected were compared to a WAG base case adjusted using standard commercial tools according to best practice. The results obtained for the current reservoir highlight key performances of the optimization algorithm applied to the WAG strategy. WAG increases the recovery factor and enhance an early production (paramount to improve the NPV according to the economic model) with respect to a continuous WI. The optimization of WAG cycles allows to find the best hydrocarbon production under the constrain imposed by the facilities and the gas availability. The optimum inter-well-distance and schedule results from the compromise between a strong sweep yet avoiding an early water breakthrough and reducing interference among producers. The well layout is therefore an output of the NPV optimization, the carbonate geological characterization and the operation risk. The optimal field development plan has shown a 35% of increment in NPV compared to the base case.This workflow presented and discussed presents an innovative and general formulation for the optimization of a WAG and/or continuous injection for real reservoir under uncertainty. The optimization algorithm allows fast convergence and accurate results. The extension to multiple objective functions and the inclusion of the human decision in the optimization process is also considered.

In contrast to most common methods used in optimal control of reservoir systems requiring a large number of decision variables, parametric rule can make a radical reduction in the number of...