Approximation Of Objective Function Research Articles

An activity level based surrogate-assisted evolutionary algorithm for many-objective optimization

Addressing expensive many-objective optimization problems (MaOPs) is a formidable challenge owing to their intricate objective spaces and high computational demands. Surrogate-assisted evolutionary algorithms (SAEAs) have gained prominence because of their ability to tackle MaOPs efficiently. They achieve this by using surrogate models to approximate objective functions, significantly reducing their reliance on costly evaluations. However, the effectiveness of many SAEAs is hampered by their reliance on various surrogate models and optimization strategies, which often result in suboptimal prediction accuracy and optimization performance. This study introduces a novel approach: an activity level based surrogate-assisted reference vector guided evolutionary algorithm specifically designed for expensive MaOPs. Utilizing the Kriging model and an angle penalty distance criterion, this algorithm effectively filters solutions that require evaluation using the original function. It employs a fixed number of training sets,that are updated via a two-screening strategy that leverages activity levels to refine population screening. This process ensures that the reference vector progressively aligns more closely with the Pareto fronts,which is enhanced by the deployment of adjusted adaptive reference vectors, thereby improving the screening precision. The proposed algorithm was tested against six contemporary algorithms using the DTLZ, WFG, and MaF test suites.The experimental results show that the proposed method outperforms other algorithms in most problems. Furthermore, its application to the cloud computing task scheduling problem underscores its practical value, demonstrating its notable effectiveness. The experimental outcomes attest to the robust performance of the algorithm across both test scenarios and real-world applications.

Minibatch Stochastic Three Points Method for Unconstrained Smooth Minimization

We present a new zero-order optimization method called Minibatch Stochastic Three Points (MiSTP), specifically designed to solve stochastic unconstrained minimization problems when only an approximate evaluation of the objective function is possible. MiSTP is an extension of the Stochastic Three Point Method (STP). The key innovation of MiSTP is that it selects the next point solely based on the objective function approximation, without relying on its exact evaluation. At each iteration, MiSTP generates a random search direction and compares the approximations of the objective function at the current point, the randomly generated direction and its opposite. The best of these three points is chosen as the next iterate. We analyze the worst-case complexity of MiSTP in the convex and non-convex cases and demonstrate that it matches the most accurate complexity bounds known in the literature for zero-order optimization methods. We perform extensive numerical evaluations to assess the computational efficiency of MiSTP and compare its performance to other state-of-the-art methods by testing it on several machine learning tasks. The results show that MiSTP outperforms or has comparable performance against state-of-the-art methods indicating its potential for a wide range of practical applications.

Multiobservation-Multichannel-Attribute-Based Multiuser Authentication for Industrial Wireless Edge Networks

In order to truly promote the further development of the Industrial Internet of Things (IIoT), terminal authentication of the IIoT is essential. Physical-Layer Authentication (PLA) has recently attracted much attention for its high security and lightweight. Nevertheless, most existing PLA schemes in conjunction only the observation of a single receiver will lead to low-reliability and low-robustness of authentication, especially in hostile time-varying wireless channels. To tackle this issue, we developed a multi-observation-multi-channel-attribute (MOMCA)-based multiuser authentication architecture, which considers both the observations of multi-receivers and multiple channel attributes of each observation to enhance wireless security. Specifically, the proposed architecture can provide additional spatial recognition characteristics for multi-users. To better fit the channel features of multi-observations, we proposed two Gradient Boosting Optimization (GBO)-based schemes. One uses Taylor expansion to approximate objective functions and adds the regularization term to avoid over-fitting issues. The other can obtain higher authentication performance by sampling the signal data with small gradient characteristics. The simulations on real industrial indoor and outdoor datasets verify the superiority of the proposed schemes in authentication accuracy over six baseline authentication schemes

Mechanic Maker 2.0: Reinforcement Learning for Evaluating Generated Rules

Automated game design (AGD), the study of automatically generating game rules, has a long history in technical games research. AGD approaches generally rely on approximations of human play, either objective functions or AI agents. Despite this, the majority of these approximators are static, meaning they do not reflect human player's ability to learn and improve in a game. In this paper, we investigate the application of Reinforcement Learning (RL) as an approximator for human play for rule generation. We recreate the classic AGD environment Mechanic Maker in Unity as a new, open-source rule generation framework. Our results demonstrate that RL produces distinct sets of rules from an A* agent baseline, which may be more usable by humans.

Using physical property surrogate models to perform accelerated multi-fidelity optimization of force field parameters.

Accurate representations of van der Waals dispersion-repulsion interactions play an important role in high-quality molecular dynamics simulations. Training the force field parameters used in the Lennard Jones (LJ) potential typically used to represent these interactions is challenging, generally requiring adjustment based on simulations of macroscopic physical properties. The large computational expense of these simulations, especially when many parameters must be trained simultaneously, limits the size of training data set and number of optimization steps that can be taken, often requiring modelers to perform optimizations within a local parameter region. To allow for more global LJ parameter optimization against large training sets, we introduce a multi-fidelity optimization technique which uses Gaussian process surrogate modeling to build inexpensive models of physical properties as a function of LJ parameters. This approach allows for fast evaluation of approximate objective functions, greatly accelerating searches over parameter space and enabling the use of optimization algorithms capable of searching more globally. In this study, we use an iterative framework which performs global optimization with differential evolution at the surrogate level, followed by validation at the simulation level and surrogate refinement. Using this technique on two previously studied training sets, containing up to 195 physical property targets, we refit a subset of the LJ parameters for the OpenFF 1.0.0 (Parsley) force field. We demonstrate that this multi-fidelity technique can find improved parameter sets compared to a purely simulation-based optimization by searching more broadly and escaping local minima. Additionally, this technique often finds significantly different parameter minima that have comparably accurate performance. In most cases, these parameter sets are transferable to other similar molecules in a test set. Our multi-fidelity technique provides a platform for rapid, more global optimization of molecular models against physical properties, as well as a number of opportunities for further refinement of the technique.

Secure Transmission for Hierarchical Information Accessibility in Downlink MU-MIMO

Physical layer security is a useful tool to prevent illegal wiretapping to confidential information. In this paper, we consider a generalized model of conventional physical layer security, referred as hierarchical information accessibility (HIA). A main feature of the HIA model is that a network has a hierarchy in information access, wherein decoding feasibility is determined by each user’s priority. Under this HIA model, we formulate a sum secrecy rate maximization problem with regard to precoding vectors. This problem is challenging since multiple non-smooth functions are involved into the secrecy rate to fulfill the HIA conditions and also the problem is non-convex. To address the challenges, we approximate the minimum function by using the LogSumExp technique, thereafter obtain the first-order optimality condition. One key observation is that the derived condition is cast as a functional eigenvalue problem, where the eigenvalue is equivalent to the approximated objective function of the formulated problem. Accordingly, we show that finding a principal eigenvector is equivalent to finding a local optimal solution. To this end, we develop a novel method called generalized power iteration for HIA (GPI-HIA). Simulations demonstrate that the GPI-HIA significantly outperforms other baseline methods in terms of the secrecy rate.

Robust design of service systems with immobile servers, general arrival and service patterns, and demand uncertainty

This paper addresses the problem of robustly designing a service system consisting of immobile servers, each modelled as a G/G/1 queuing system, when the arrival rates are not known with certainty. The problem involves locating service centers, determining their capacities and assigning customers to them to minimize the total cost, which includes the setup, access and waiting costs. Besides the nominal problem, two robust problems with budget and ball uncertainty sets are considered. A piecewise-linear approximation is applied to handle the nonlinear waiting cost, which enables all the problems to be tightly approximated as mixed-integer quadratic programs. We also propose a Lagrangian approach that is capable of finding high-quality solutions and strong bounds for instances of practical sizes. Numerical experiments were conducted to validate the proposed models and solution methods and to study the effect of the problem parameters, the uncertainty set size and the objective function approximations on the optimal solution.

Optimization on Multiuser Physical Layer Security of Intelligent Reflecting Surface-Aided VLC

This letter investigates physical layer security in intelligent reflecting surface (IRS)-aided VL communication (VLC). Under the point source assumption, we first elaborate the system model in the scenario with multiple legitimate users and one eavesdropper, where the secrecy rate maximization problem is transformed into an assignment problem by objective function approximation. Then, an iterative Kuhn-Munkres algorithm is proposed to optimize the transformed problem, and its computational complexity is in the second-order form of the numbers of IRS units and transmitters. Moreover, numerical simulations are carried out to verify the approximation performance and the VLC secrecy rate improvement by IRS.

A noniterative solution to the inverse Ising problem using a convex upper bound on the partition function

The inverse Ising problem, or the learning of Ising models, is notoriously difficult, as evaluating the partition function has a large computational cost. To quickly solve this problem, inverse formulas using approximation methods such as the Bethe approximation have been developed. In this paper, we employ the tree-reweighted (TRW) approximation to construct a new inverse formula. An advantage of using the TRW approximation is that it provides a rigorous upper bound on the partition function, allowing us to optimize a lower bound for the learning objective function. We show that the moment-matching and self-consistency conditions can be solved analytically, and we obtain an analytic form of the approximate interaction matrix as a function of the given data statistics. Using this solution, we can compute the interaction matrix that is optimal to the approximate objective function without iterative computation. To evaluate the accuracy of the derived learning formula, we compared our formula to those obtained by other approximations. From our experiments on reconstructing interaction matrices, we found that the proposed formula gives the best estimates in models with strongly attractive interactions on various graphs.

Approximation Method for Optimization Problems in Gate-Model Quantum Computers

In near-term quantum computers, the computations are realized via unitary operators. The optimization problem fed into the quantum computer sets an objective function that is to be estimated via several measurement rounds. Here, we define a procedure for objective function approximation in gate-model quantum computers. The proposed solution optimizes the process of objective function estimation for optimization problems in gate-model quantum computers and quantum devices.

Learning performance of regularized regression with multiscale kernels based on Markov observations

We analyze a least square regularized regression (LSRR) problem with multiscale kernels on the assumption that observations are subject to be non-independent and identically distributed, such as uniformly ergodic Markov chain (u.e.M.c.) observations. Unlike some existing known analysis for non-i.i.d. observations, we establish error bound on the learning performance according to the complexity of hypothesis spaces with the u.e.M.c. observations, and also derive the learning rate of the multiscale LSRR (i.e., MLSRR) problem with the u.e.M.c. observations, which is with the order of O(m−1). The Markov observing algorithm with MLSRR has also been proposed to generate the u.e.M.c. observations for handling nonflat objective function approximation problem, and then we provide empirical evaluations on simulation dataset and UCI repository to compare the learning performance of MLSRR algorithm with u.e.M.c. observations and i.i.d. observations.

BUILDING STRUCTURE OPTIMIZATION BASED ON A HIGH-DIMENSIONAL MODEL REPRESENTATION APPROACH

In this study, a model reduction technique based on a high-dimensional model representation (HDMR) approach was investigated and applied to design optimization of building structures. Those structures have long been designed using engineering intuition and an iterative trial-and-error method. In order to evaluate structural responses, a finite element (FE) analysis code is generally required. Gradient-based numerical algorithms and evolutionary algorithms are widely available and can be adopted to the design optimization of structures. An alternative category of optimization methods relies on approximate objective or constraint functions that can be created using various interpolation or regression techniques. In this work, the model reduction was achieved using augmented radial basis functions (RBFs) as component functions of HDMR. After sample points were generated along each variable axis, detailed FE analyses were conducted to evaluate building responses, which were used for constructing RBF-HDMR models of structural responses. The optimization was performed using a standard gradient-based numerical method. The accuracy of the RBF-HDMR could be improved if the optimal design point was added as an additional sample point. One advantage of the proposed optimization approach was that the interface programming with any existing FE code was not necessary. To illustrative the application of the method, a high-rise building was studied and optimized in order to reduce the building’s global torsional responses. The proposed optimization method worked well for the example.

Non-convex ℓ p regularization for sparse reconstruction of electrical impedance tomography

This work is to investigate the image reconstruction of electrical impedance tomography from the electrical measurements made on an object's surface. An -norm (0<p<1) sparsity-promoting regularization is considered to deal with the fully non-linear electrical impedance tomography problem, and a novel type of smoothing gradient-type iteration scheme is introduced. To avoid the difficulty in calculating its gradient in the optimization process, a smoothing Huber potential function is utilized to approximate the -norm penalty. We then propose the smoothing algorithm in the general frame and establish that any accumulation point of the generated iteration sequence is a first-order stationary point of the original problem. Furthermore, one iteration scheme based on the homotopy perturbation technology is derived to find the minimizers of the Huberized approximated objective function. Numerical experiments show that non-convex -norm sparsity-promoting regularization improves the spatial resolution and is more robust with respect to noise, in comparison with -norm regularization.

Optimization of the shape of regions supporting boundary conditions

This article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a ‘physical’ partial differential equation. At first, we analyze a model situation involving the solution $$u_\varOmega $$ to a Laplace equation in a domain $$\varOmega $$ ; the boundary $$\partial \varOmega $$ is divided into three parts $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ , supporting respectively homogeneous Dirichlet, homogeneous Neumann and inhomogeneous Neumann boundary conditions. The shape derivative $$J^\prime (\varOmega )(\theta )$$ of a general objective function $$J(\varOmega )$$ of the domain is calculated in the framework of Hadamard’s method when the considered deformations $$\theta $$ are allowed to modify the geometry of $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ (i.e. $$\theta $$ does not vanish on the boundary of these regions). The structure of this shape derivative turns out to depend very much on the regularity of $$u_\varOmega $$ near the boundaries of the regions $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ . For this reason, in particular, $$J^\prime (\varOmega )$$ is difficult to calculate and to evaluate numerically when the transition $$\overline{\varGamma _D} \cap {\overline{\varGamma }}$$ between homogeneous Dirichlet and homogeneous Neumann boundary conditions is subject to optimization. To overcome this difficulty, an approximation method is proposed, in which the considered ‘exact’ Laplace equation with mixed boundary conditions is replaced with a ‘smoothed’ version, featuring Robin boundary conditions on the whole boundary $$\partial \varOmega $$ with coefficients depending on a small parameter $$\varepsilon $$ . We prove the consistency of this approach in our model context: the approximate objective function $$J_\varepsilon (\varOmega )$$ and its shape derivative converge to their exact counterparts as $$\varepsilon $$ vanishes. Although it is rigorously justified only in a model problem, this approximation methodology may be adapted to many more complex situations, for example in three space dimensions, or in the context of the linearized elasticity system. Various numerical examples are eventually presented in order to appraise the efficiency of the proposed approximation process.

A new local update-based method for reliability-based design optimization

Reliability-based design optimization (RBDO) is often associated with computationally expensive computer simulations in practical engineering applications, in which both objective and constraint functions may be implicit and time-consuming to evaluate. To improve the efficiency of RBDO, metamodels are often employed to substitute the implicit and expensive functions. In this paper, a new local update-based method is proposed for RBDO under the approximation of both objective and constraint functions. In this method, Kriging metamodels are employed to replace both objective and constraint functions, and sequential optimization and reliability assessment (SORA) is adopted to perform RBDO. Meanwhile, a local update method is developed for the approximation of objective and constraint functions, termed as LUOC. In LUOC, two local update strategies are proposed. For the constraint functions, one local update strategy judges the active constraints and then sequentially refines their Kriging metamodels in the local regions around the most probable target points obtained by SORA. For the objective function, the other local update strategy based on expected improvement is developed to refine the Kriging metamodel in the local region around the SORA solution with considering the Kriging prediction variance. Three examples including a battery design application are presented to test the accuracy and efficiency of the proposed method for RBDO.

User-Centric Joint Access-Backhaul Design for Full-Duplex Self-Backhauled Wireless Networks

Full-duplex self-backhauling is promising to provide cost-effective and flexible backhaul connectivity for ultra-dense wireless networks, but also poses a great challenge to resource management between the access and backhaul links. In this paper, we propose a user-centric joint access-backhaul transmission framework for full-duplex self-backhauled wireless networks. In the access link, user-centric clustering is adopted so that each user is cooperatively served by multiple small base stations (SBSs). In the backhaul link, user-centric multicast transmission is proposed so that each user’s message is treated as a common message and multicast to its serving SBS cluster. We first formulate an optimization problem to maximize the network weighted sum rate through joint access-backhaul beamforming and SBS clustering when global channel state information (CSI) is available. This problem is efficiently solved via the successive lower-bound maximization approach with a novel approximate objective function and the iterative link removal technique. We then extend the study to the stochastic joint access-backhaul beamforming optimization with partial CSI. Simulation results demonstrate the effectiveness of the proposed algorithms for both full CSI and partial CSI scenarios. They also show that the transmission design with partial CSI can greatly reduce the CSI overhead with little performance degradation.

QUEST – A new quadratic decision model for the multi-satellite scheduling problem

Demand for earth observation satellite imagery is pervasive and increasing rapidly across multiple domains, highlighting the key problem to optimally schedule satellite image acquisition/collection, subject to ground and on-board constraints. Despite the variety of reported approaches to solving the NP-hard multi-satellite scheduling problem (m-SatSP), serious limitations still prevail, largely overlooking problem structure exploitation and/or domain knowledge. Assuming a few constraints, problem modelling—or scope—is often confined to a simple trailing satellite constellation composition perspective and tends to conveniently oversimplify footprint coverage intricacies. As a result, adequate problem task decomposition properly reflecting complex kinematic behaviour for an ad hoc satellite constellation remains elusive. Known approaches also fail to provide valuable solution optimality gap estimations to objectively qualify best computed solution and/or to control run-time execution in solving hard problem instances. In this paper, a novel approach to solving the single objective static m-SatSP is proposed. It is based on network flow optimization using mathematical programming. Unlike previous methods, QUEST (QUadratically constrainEd program Solver Technology) relies on a sound alternative approximate objective function and, the exploitation of exact problem-solving techniques. Derived from domain knowledge and problem structure considerations, QUEST generalizes problem modelling to successfully handle virtual constellation, avoiding unsuitable utilization of traditional area coverage decomposition scheme. The proposed decision model concurrently captures coverage approximation, imaging success uncertainty and quality for a variety of tasks. It also includes new and optional constraints while embracing an acceptable upper bound on collection value. A QUEST variant alternatively relying on “delayed reward” to bridge promising search regions on move selection, further shows optimality gap reduction and provides additional speedup. Computational results prove QUEST to be cost-effective and to outperform some recent baseline methods derived from best-known m-SatSP procedures. It comparatively demonstrates measurable collection and run-time gains, and provides a tight upper bound on the optimal solution of hard problems.

Automatic Spatial Calibration of Ultra-Low-Field MRI for High-Accuracy Hybrid MEG-MRI.

With a hybrid magnetoencephalography (MEG)-MRI device that uses the same sensors for both modalities, the co-registration of MRI and MEG data can be replaced by an automatic calibration step. Based on the highly accurate signal model of ultra-low-field (ULF) MRI, we introduce a calibration method that eliminates the error sources of traditional co-registration. The signal model includes complex sensitivity profiles of the superconducting pickup coils. In the ULF MRI, the profiles are independent of the sample and therefore well-defined. In the most basic form, the spatial information of the profiles, captured in parallel ULF-MR acquisitions, is used to find the exact coordinate transformation required. We assessed our calibration method by simulations assuming a helmet-shaped pickup-coil-array geometry. Using a carefully constructed objective function and sufficient approximations, even with low-SNR images, sub-voxel and sub-millimeter calibration accuracy were achieved. After the calibration, distortion-free MRI and high spatial accuracy for MEG source localization can be achieved. For an accurate sensor-array geometry, the co-registration and associated errors are eliminated, and the positional error can be reduced to a negligible level.

Explicit model of dual programming and solving method for a class of separable convex programming problems

ABSTRACTAn objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.

Redundancy optimization of cold-standby systems under periodic inspection and maintenance

In this paper, a redundancy allocation problem for cold-standby systems with degrading components is studied. We consider performing a policy of periodic inspection and preventive maintenance on the cold-standby components. Components in cold-standby mode are assumed to suffer from performance degradation when exposed in extreme standby environments for a long-term storage. Except the online working components, other cold-standby components are periodically inspected and preventively maintained to retain them in or restore them to an acceptable condition. The situation, that the cold-standby components are restored like new but their failure rates increase, is considered. In this paper, we develop a model to determine an optimal combination of the component choices, the number of redundant components, and the length of periodic inspection interval, so as to maximize the system reliability while meeting the resource limits, such as cost and weight. As it is difficult to obtain a closed-form expression of the objective function for the optimization problem, an approximated objective function is proposed. The approximation is also used for evaluating the expected cost of periodic inspection and maintenance. We then develop a modified genetic algorithm to solve the optimization problem. Finally, a numerical example is presented to illustrate the effectiveness of the proposed model and the solution methodology.