Dimensional Optimization Problem Research Articles

Solving higher dimensional expensive black box global optimization problems using sparse directional search on surrogates

We present a new algorithm Directional Optimization Search with Surrogate (DOSS), for optimizing problems with box constraints and a computationally expensive black-box objective function. DOSS is a radial basis function (RBF) based method that mainly focuses on higher dimensional and computationally expensive objective functions that can be multimodal. DOSS introduces three new techniques not previously used in earlier RBF algorithms, including using a combination of the coordinates knowledge level, fewer initial sampling points, and the surrogate’s gradient information. Numerical results on a test set including 14 test problems with 36, 48, and 60 dimensions show that DOSS outperforms two recently published algorithms RBFOpt and TuRBO, and earlier RBF algorithms such as DYCORS. TuRBO is a Gaussian process based optimization algorithm, which outperformed earlier state-of-the-art methods. DOSS algorithm also has a good performance on real-world optimization problems, including robot pushing and rover trajectory planning problems. Almost sure convergence for the DOSS algorithm is also proven in this paper. An implementation of DOSS is available at https://doi.org/10.5281/zenodo.13731558.

Multi-input Fourier neural network and its sparrow search optimization

In engineering applications, the back-propagation (BP) neural network often encounters many limitations due to its slow convergence and high noise sensitivity, and meanwhile the reported Fourier neural networks have no ability to extract the features of multi-attribute input data. Hereby, This work proposes a gradient descent-based multi-input Fourier neural network after integrating the multi-layer perceptron with an overlapping Fourier neural network. Thereafter, related to the difficulty of deciding the global optimal parameter settings, an improved sparrow search algorithm is developed to optimize the parameter settings and solve high dimensional function optimization problems, after the Cat chaotic map and the mechanisms of population-size adjustment and parameter adaptiveness are designed to promote the sparrow search algorithm's ability to balance global exploration and local exploitation. The theoretical analysis shows that the improved algorithm's computational complexity is decided by its population size and the optimization problem's dimension. Numerically comparative experiments have validated that not only the acquired Fourier neural network can effectively extract the features of multi-attribute data with strong generalization ability, but also the improved algorithm has significant advantages in coping with high dimensional function optimization problems.

Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

Surrogate-assisted differential evolution using manifold learning-based sampling for high- dimensional expensive constrained optimization problems

To address the challenges of high-dimensional constrained optimization problems with expensive simulation models, a Surrogate-Assisted Differential Evolution using Manifold Learning-based Sampling (SADE-MLS) is proposed. In SADE-MLS, differential evolution operators are executed to generate numerous high-dimensional candidate points. To alleviate the curse of dimensionality, a Manifold Learning-based Sampling (MLS) mechanism is developed to explore the high-dimensional design space effectively. In MLS, the intrinsic dimensionality of the candidate points is determined by a maximum likelihood estimator. Then, the candidate points are mapped into a low-dimensional space using the dimensionality reduction technique, which can avoid significant information loss during dimensionality reduction. Thus, Kriging surrogates are constructed in the low-dimensional space to predict the responses of the mapped candidate points. The candidate points with high constrained expected improvement values are selected for global exploration. Moreover, the local search process assisted by radial basis function and differential evolution is performed to exploit the design space efficiently. Several numerical benchmarks are tested to compare SADE-MLS with other algorithms. Finally, SADE-MLS is successfully applied to a solid rocket motor multidisciplinary optimization problem and a re-entry vehicle aerodynamic optimization problem, with the total impulse and lift to drag ratio being increased by 32.7% and 35.5%, respectively. The optimization results demonstrate the practicality and effectiveness of the proposed method in real engineering practices.

Shape optimization for the Stokes system with threshold leak boundary conditions

This paper discusses the process of optimizing the shape of systems that are controlled by the Stokes flow with threshold leak boundary conditions. In the theoretical part it focuses on studying the stability of solutions to the state problem in relation to a specific set of domains. In order to facilitate computation, the slip term and impermeability condition are regulated. In the computational part, the optimized portion of the boundary is defined using Bézier polynomials, in order to create a finite dimensional optimization problem. The paper also includes numerical examples to demonstrate the computational efficiency of this approach.

Football Optimization Algorithm (FbOA): A Novel Metaheuristic Inspired by Team Strategy Dynamics

The Football Optimization Algorithm (FbOA) is introduced as a novel population-based metaheuristic optimization technique inspired by the dynamic strategies of a football team. Designed to address complex optimization problems characterized by high dimensionality, nonlinearity, and multiple local optima, FbOA draws on the strategic balance between exploration and exploitation observed in football gameplay. The algorithm mimics players’ tactical positioning and movement, incorporating short passes, long passes, and positional adjustments to explore and exploit the solution space effectively. This study comprehensively evaluates the performance of FbOA using benchmark functions from the CEC 2005 test suite with 30-dimensional and 100- dimensional optimization problems. The results demonstrate that FbOA outperforms several state-of-the-art metaheuristic algorithms regarding convergence speed, accuracy, and robustness. The findings suggest that FbOA offers a promising alternative for solving various optimization challenges across multiple fields.

The design and optimization of heterogeneous catalysts using computational methods

Computational design of catalytic materials is a high dimensional structure optimization problem that is limited by the bottleneck of expensive quantum computation tools. An illustration of interaction of different factors involved in the design and optimization of a catalyst.

A time independent least squares algorithm for parameter identification of Turing patterns in reaction-diffusion systems.

Turing patterns arising from reaction-diffusion systems such as epidemic, ecology or chemical reaction models are an important dynamic property. Parameter identification of Turing patterns in spatial continuous and networked reaction-diffusion systems is an interesting and challenging inverse problem. The existing algorithms require huge account operations and resources. These drawbacks are amplified when apply them to reaction-diffusion systems on large-scale complex networks. To overcome these shortcomings, we present a new least squares algorithm which is rooted in the fact that Turing patterns are the stationary solutions of reaction-diffusion systems. The new algorithm is time independent, it translates the parameter identification problem into a low dimensional optimization problem even a low order linear algebra equations. The numerical simulations demonstrate that our algorithm has good effectiveness, robustness as well as performance.

Pair barracuda swarm optimization algorithm: a natural-inspired metaheuristic method for high dimensional optimization problems

High-dimensional optimization presents a novel challenge within the realm of intelligent computing, necessitating innovative approaches. When tackling high-dimensional spaces, traditional evolutionary tools often encounter pitfalls, including dimensional catastrophes and a propensity to become trapped in local optima, ultimately compromising result accuracy. To address this issue, we introduce the Pair Barracuda Swarm Optimization (PBSO) algorithm in this paper. PBSO employs a unique strategy for constructing barracuda pairs, effectively mitigating the challenges posed by high dimensionality. Furthermore, we enhance global search capabilities by incorporating a support barracuda alongside the leading barracuda pair. To assess the algorithm’s performance, we conduct experiments utilizing the CEC2017 standard function and compare PBSO against five state-of-the-art natural-inspired optimizers in the control group. Across 29 test functions, PBSO consistently secures top rankings with 9 first-place, 13 second-place, 5 third-place, 1 fourth-place, and 1 fifth-place finishes, yielding an average rank of 2.0345. These empirical findings affirm that PBSO stands as the superior choice among all test algorithms, offering a dependable solution for high-dimensional optimization challenges.

The Competition Complexity of Dynamic Pricing

We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward [Formula: see text] achievable by the optimal online policy on m independent and identically distributed (i.i.d.) random variables distributed according to F to the expected maximum [Formula: see text] of n i.i.d. draws from F. We ask how big m has to be to ensure that [Formula: see text] for all F. We resolve this question and characterize the competition complexity as a function of ε. When [Formula: see text], the competition complexity is unbounded. That is, for any n and m there is a distribution F such that [Formula: see text]. In contrast, for any [Formula: see text], it is sufficient and necessary to have [Formula: see text], where [Formula: see text]. Therefore, the competition complexity not only drops from unbounded to linear, it is actually linear with a very small constant. The technical core of our analysis is a lossless reduction to an infinite dimensional and nonlinear optimization problem that we solve optimally. A corollary of this reduction is a novel proof of the factor [Formula: see text] i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds. Funding: This work was supported by ANID (Anillo ICMD) [Grant ACT210005] and the Center for Mathematical Modeling [Grant FB210005].

Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

Subspace segmentation based co-evolutionary algorithm for balancing convergence and diversity in many-objective optimization

With the increase of the objective dimension of optimization problems, the effect of comparing individuals through Pareto dominance relation drops sharply. While some algorithms enhance Pareto dominance via diversity preservation strategies, performance indicators, and reference vectors, many of them encounter difficulties in balancing the convergence and diversity of populations. Therefore, this paper proposes a subspace segmentation based co-evolutionary algorithm for balancing convergence and diversity in many-objective optimization. First, the decision space is divided into a convergence subspace and a diversity subspace, which are searched in the early and late stages to improve the population convergence and diversity, respectively. Second, a capacity adaptively adjusted archive is used to retain elite individuals with better convergence in the population, which is further used to mate with the population. Moreover, an indicator with penalty factor is proposed to retain the boundary individuals so as to maintain the population diversity. Comparing with 6 advanced many-objective evolutionary algorithms on 63 benchmark cases, the proposed algorithm obtains smallest IGD on 36 benchmark cases, the experimental results show that the proposed algorithm can balance convergence and diversity well and has exhibit competitiveness.

A decision-making design approach to optimize the hole arrangement of effusion cooling plates under non-uniform heat load conditions

The harsh operating conditions of aero-engines contain non-uniform temperatures and variable hotspots, which require adaptive cooling techniques to accommodate the heat loads on the external surfaces of metal. Among the existing technologies, effusion cooling is an important way to fulfill such quests due to its flexibility in controlling coolant locally. As an effusively cooled component generally contains hundreds of holes, hole arrangement optimization is a high-dimensional problem that is difficult to solve using existing evolutionary algorithms. Recently matured deep reinforcement learning algorithms showed success in many research fields by incorporating data modeling with sequential optimization, which demonstrated the ability to convert high dimensional optimization problems into very low dimensional iterative optimization problems. The present study proposed a decision-making design approach to optimize the hole arrangement of a series of effusion cooling plates. Hole arrangements were transformed into Markov decision chains of opening or closing each hole. The decision chains were solved by a reinforcement learning framework integrated with a deep learning-based environment simulator. Results indicated that the proposed approach could yield an elevation of cooling performance by 20% as compared with the results from genetic algorithms. Moreover, an analytical correlation was extracted from the clear decision boundaries of reinforcement learning. Upon further validation, the computational cost to apply the correlation to other non-uniform heat loads was within seconds, while the cooling effect could reach the same optimization level as reinforcement learning. The idea proposed in this study could be applied to the arrangement optimization of multiple cooling technologies, such as film/effusion cooling, impingement cooling, and pin–fin array cooling.

A power quality‐based framework for optimal siting and sizing of wind turbines by search space reduction via power losses and flicker emission sensitivity indices

AbstractAs a well‐established renewable energy resource, wind energy is being widely used in distribution networks. However, flicker emission produced by wind turbines (WTs) is one of the main disadvantages which may cause serious problems during the system operation. Power quality‐based optimal allocation (siting and sizing) of WTs is an appropriate way to mitigate the flicker emission. Since the planning issue is a high dimensional and complex optimization problem and optimization methods suffer from premature convergence, search space reduction may be an effective approach to optimally solve the allocation problem. Here, to determine the importance of the network buses, two indices, namely, power loss index (PLI) and flicker emission index (FEI) are introduced to determine the sensitivity of the network losses and flicker emission to allocate WT on each bus. Then, based on the proposed indices, a number of important buses are identified and the allocation problem is solved with respect to these buses. Over the case study, it can be drawn that optimal allocation of WTs considerably affects the flicker emission. Moreover, search space reduction is an efficient way to optimally solve the allocation problem.

Cloud-Based Computational Model Predictive Control Using a Parallel Multiblock ADMM Approach

Heavy computational load for solving nonconvex problems for large-scale systems or systems with real-time demands at each sample step has been recognized as one of the reasons for preventing a wider application of nonlinear model predictive control (NMPC). To improve the real-time feasibility of NMPC with input nonlinearity, we devise an innovative scheme called cloud-based computational model predictive control (MPC) by using an elaborately designed parallel multi-block alternating direction method of multipliers (ADMM) algorithm. This novel parallel multi-block ADMM algorithm is tailored to tackle the computational issue of solving a nonconvex problem with nonlinear constraints. It is ensured that the designed algorithm converges to a locally optimal solution of the optimization problem under reasonable assumptions by using the Kurdyka–Łojasiewicz property. With the help of this distributed optimization algorithm, a computational MPC scheme is developed, which can transform the NMPC optimization problem into a set of subproblems only associated with the decision variables at one prediction step. Through the parallel computing algorithm, the computational MPC can deal with large computational loads caused by high dimensional optimization problems, and improve computational efficiency. Furthermore, to allow for a more efficient implementation of the developed computational MPC and alleviate local calculation loads, a cloud-based computational MPC architecture is devised, which makes significantly better use of computational resources provided by a cloud server. An important advantage of this architecture with Docker (https://www.docker.com/) container to implement parallelization is that it does not lead to large increases in the solution time regardless of how long the prediction horizon is set. Finally, the developed cloud-based computational MPC architecture is trialed on a group of plug-in hybrid electric vehicles (PHEVs).

A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. Measurability and convergence rates of a stationarity measure are handled, filling a gap for applications to nonconvex infinite dimensional stochastic optimization problems. The method is demonstrated on an optimal control problem constrained by a class of elliptic semilinear partial differential equations (PDEs) under uncertainty.

Heterogeneous Transformer: A Scale Adaptable Neural Network Architecture for Device Activity Detection

To support modern machine-type communications, a crucial task during the random access phase is device activity detection, which is to identify the active devices from a large number of potential devices based on the received signal at the access point. By utilizing the statistical properties of the channel, state-of-the-art covariance based methods have been demonstrated to achieve better activity detection performance than compressed sensing based methods. However, covariance based methods require to solve a high dimensional nonconvex optimization problem by updating the estimate of the activity status of each device sequentially. Since the number of updates is proportional to the device number, the computational complexity and delay make the iterative updates difficult for real-time implementation especially when the device number scales up. Inspired by the success of deep learning for real-time inference, this paper proposes a learning based method with a customized heterogeneous transformer architecture for device activity detection. By adopting an attention mechanism in the architecture design, the proposed method is able to extract features reflecting relevance among device pilots and received signal, permutation equivariant with respect to devices, and its training parameter number is independent of the device number. Simulation results demonstrate that the proposed method achieves better activity detection performance with much shorter computation time than state-of-the-art covariance approach, and generalizes well to different numbers of devices and BS-antennas, different pilot lengths, transmit powers, and cell radii.

Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem

This paper is concerned with a two-dimensional fractional optimal control problem whose governing equations are distributed order fractional differential equations in the Caputo sense. A generalized fractional-order Legendre wavelet method has been used to solve the two-dimensional distributed-order fractional optimal control problem. An exact formula for the Riemann–Liouville integration of generalized fractional-order Legendre wavelet has been derived by using regularized beta functions. This formula and the two-dimensional Gauss–Legendre integration formula have been used to solve the two-dimensional distributed order fractional optimal control problem. Moreover, an L2-error estimate in the approximation of an unknown function with a generalized fractional-order Legendre wavelet has been derived and the estimated order has been verified for a given function. Furthermore, convergence analysis for the proposed method has been presented. In the last, two test problems have been considered to illustrate the efficiency of the proposed method.

A Novel Variant of Moth Flame Optimizer for Higher Dimensional Optimization Problems

A Novel Variant of Moth Flame Optimizer for Higher Dimensional Optimization Problems

A convex optimization approach to multi-objective design of two-stage compensators for MIMO linear systems

The previous design of two-stage compensators with multi-objectives concerns the closed-loop stability, the low-sensitivity, and the satisfactory time-domain performance. However, the design only considers the single-input single-output system. This paper aims to present the design of two-stage compensators for multi-input multi-output (MIMO) linear systems which contain dynamical interaction between inputs and outputs. When Youla parameterization is applied to the MIMO linear system, the design criteria can be cast as a nonlinear optimization. In addition, the infinite dimension optimization problem is approximated to a finite dimension problem with Ritz approximation. The suitable form of design parameters is proposed to ensure the low sensitivity of the MIMO system. The two-stage compensator design is formulated as convex optimization which is efficiently solved using available solvers. Finally, the two-stage compensators are designed for Wood and Berry’s model of binary distillation column. The numerical results indicate that all design objectives of frequency domain and time domain can be achieved by the two-stage compensators.