Solution Of Optimization Problem Research Articles

Inventory Optimization with Several Resources

Inventory optimization is extended with additional constraints that affect production efficiency using estimated inventory. The extension of the inventory problem is in a special formal form with an additional optimization problem. The latter is aimed at minimizing production costs while using optimal volumes of material stocks. Such integration of inventory and production is formalized through a bi-level optimization problem. The case of simultaneous delivery of different types of goods is strongly related to the production process of final goods. A bi-level optimization problem is numerically defined and solved. Delivery costs are minimized for a given production plan. The bi-level problem is applied to prepare feed with the required nutrient content while minimizing the inventory costs of supplying the required raw materials. Empirical results give advantages for the obtained solution of the bi-level optimization problem.

Survey of interactive evolutionary decomposition-based multiobjective optimization methods.

Interactive methods support decision-makers in finding the most preferred solution for multiobjective optimization problems, where multiple conflicting objective functions must be optimized simultaneously. These methods let a decision-maker provide preference information iteratively during the solution process to find solutions of interest, allowing them to learn about the trade-offs in the problem and the feasibility of the preferences. Several interactive evolutionary multiobjective optimization methods have been proposed in the literature. In the evolutionary computation community, the so-called decomposition-basedmethods have been increasingly popular because of their good performance in problems with many objective functions. They decompose the multiobjective optimization problem into multiple sub-problems to be solved collaboratively. Various interactive versions of decomposition-based methods have been proposed. However, most of them do not consider the desirable properties of real interactive solution processes, such as avoiding imposing a high cognitive burden on the decision-maker, allowing them to decide when to interact with the method, and supporting them in selecting a final solution. This paper reviews interactive evolutionary decomposition-based multiobjective optimization methods and different methodologies utilized to incorporate interactivity in them. Additionally, desirable properties of interactive decomposition-based multiobjective evolutionary optimization methods are identified, aiming to make them easier to be applied in real-world problems.

A Novel Prescribed-Time Convergence Acceleration Algorithm with Time Rescaling

In machine learning, the processing of datasets is an unavoidable topic. One important approach to solving this problem is to design some corresponding algorithms so that they can eventually converge to the optimal solution of the optimization problem. Most existing acceleration algorithms exhibit asymptotic convergence. In order to ensure that the optimization problem converges to the optimal solution within the prescribed time, a novel prescribed-time convergence acceleration algorithm with time rescaling is presented in this paper. Two prescribed-time acceleration algorithms are constructed by introducing time rescaling, and the acceleration algorithms are used to solve unconstrained optimization problems and optimization problems containing equation constraints. Some important theorems are given, and the convergence of the acceleration algorithms is proven using the Lyapunov function method. Finally, we provide numerical simulations to verify the effectiveness and rationality of our theoretical results.

INNOVATIVE IMPROVED APPROXIMATE SOLUTION METHOD FOR THE INTEGER KNAPSACK PROBLEM, ERROR COMPRESSION AND COMPUTATIONAL EXPERIMENTS

Context. Mathematical models of many optimization problems encountered in economics and engineering are taken in the form of an integer knapsack problem. Since this problem belongs to the class of “NP-complete”, that is, “hard to solve” problems, the number of operations required by known methods to find its optimal solution is exponential. This does not allow solving large-scale problems in real time. Therefore, various and fast working approximate solution methods of this problem have been developed. However, it is known that the approximate solution provided by those methods can differ significantly from the optimal solution in most cases. Therefore, after taking any approximate solution as a starting point, there is a demand to develop methods for its further improvement. Development of such methods has both theoretical and great practical importance. Objective. The main purpose solving of this issue is as follows. The main purpose in performing this work is to first find an initial approximate solution of the problem using any known method, and then work out an algorithm for successively further improvement of this solution. For this purpose, the set of numbers with which the coordinates of the optimal solution and the found approximate solution can differ should be determined. After that, new solutions should be constructed by assigning possible values to the unknowns corresponding to the numbers in that set, and the best among these solutions should be selected. However, the algorithm for constructing such a solution should be simple, require a small number of operations, not cause difficulties from the point of view of programming, be new and be applicable to practical issues. Method. The essence of the proposed method consists of the following. First, the initial approximate solution of the considered problem and the value of the objective function corresponding to this solution are found by a known rule. After that, the optimal solution of the problem is easily found by a known method, without taking into account the condition that the unknowns are integers. Obviously, this solution can take at most one coordinate fractional value. It is assumed that the coordinates of the optimal solution of the integer knapsack problem and the initial approximate solution may differ around a certain fractional coordinate of the optimal solution of the continuous problem. Then, the minimum number of non-zero coordinates and zero coordinates in the optimal solution is found. Corresponding theorems have been proved for this. It is assumed that the different coordinates of the optimal solution and the initial approximate solution located between those minimal numbers. Therefore, the best solution can be selected by successively changing the coordinates between those minimum numbers one by one. Results. Extensive calculation experiments were conducted with the application of the proposed method.To have a high quality of this method was confirmed once again through experiments. Conclusions. The proposed method is new, simple in nature, easy to consider from the programming point of view, and has important practical importance. Thus, we call this solution the innovative improved approximate solution.

Neighborhood persistency of the linear optimization relaxation of integer linear optimization

AbstractFor an integer linear optimization (ILO) problem, persistency of its linear optimization (LO) relaxation is a property that for every optimal solution of the relaxation that assigns integer values to some variables, there exists an optimal solution of the ILO problem in which these variables retain the same values. Although persistency has been used to develop heuristic, approximation, and fixed-parameter algorithms for special cases of ILO, its applicability remains unknown in the literature. In this paper, we reveal a maximal subclass of ILO such that its LO relaxation has persistency. Specifically, we show that the LO relaxation of ILO on unit-two-variable-per-inequality (UTVPI) systems has persistency and is (in a certain sense) maximal among such ILO. Our result generalizes the persistency results by Nemhauser and Trotter, Hochbaum et al., and Fiorini et al. Even more, we propose a stronger property called neighborhood persistency and show that the LO relaxation of ILO on UTVPI systems in general has this property. Using this stronger result, we obtain a fixed-parameter algorithm (where the parameter is the optimal value) and another proof of 2-approximability for ILO on UTVPI systems where objective functions and variables are non-negative.

Effectively encoding satisfiability problems into Ising models for quantum annealing

Ising Models, defined by quadratic objective functions (or Hamiltonians), enable to use quantum annealers to search for optimal or near-optimal solutions of satisfiability problems. However, current quantum annealers have limited resolution, meaning that small or closely-valued coefficients in the Hamiltonian may be obscured by flux noise, leading to a degradation in the performance of quantum annealing. In this paper, we propose a novel design methodology for encoding satisfiability problems into Ising models via Integer Linear Programming. Experimental results show that our method can effectively reduce the resolution requirements for quantum annealers.

Multi-Objective Optimization of a Two-stage Helical Gearbox with Double Gears in the First Stage using MARCOS

This study demonstrates the solution of the Multi-Objective Optimization Problem (MOOP) of a two-stage helical gearbox with double gears at the first stage, following the MARCOS methodology. The goal of this work is to identify the most effective essential design factors to reduce the bottom area of the gearbox while maximizing its efficiency, which constituted a significant novel finding. For this purpose, three crucial design parameters were selected, the first stage gear ratio and the wheel face width (Xba) coefficients for the first and second stage. Furthermore, the Multi-Criteria Decision Making (MCDM) issue was chosen to be handled by the MARCOS method, and the weight criterion for solving the MOOP was determined by the MEREC method. The drawn conclusions are useful in developing a two-stage helical gearbox with double gears at the first stage by helping to identify the ideal values for the three important design parameters.

Coevolutionary Neural Solution for Nonconvex Optimization With Noise Tolerance.

The existing solutions for nonconvex optimization problems show satisfactory performance in noise-free scenarios. However, they are prone to yield inaccurate results in the presence of noise in real-world problems, which may lead to failures in optimizing nonconvex problems. To this end, in this article, we propose a coevolutionary neural solution (CNS) by combining a simplified neurodynamics (SND) model with the particle swarm optimization (PSO) algorithm. Specifically, the proposed SND model does not leverage the time-derivative information, exhibiting greater stability compared to existing models. Furthermore, due to the noise tolerance capacity and rapid convergence property exhibited by the SND model, the CNS can rapidly achieve the optimal solution even in the presence of various perturbations. Theoretical analyses ensure that the proposed CNS is globally convergent with robustness and probability. In addition, the effectiveness of the CNS is compared with those of the existing solutions by a class of illustrative examples. We further apply the proposed solution to design a finite impulse response (FIR) filter and a pressure vessel to demonstrate its performance.

A robust seismic wavefield modeling method based on minimizing spatial simulation error using L2-norm cost function

To reduce the spatial simulation error generated by the finite difference method, previous researchers compute the optimal finite-difference weights always by minimizing the error of spatial dispersion relation. However, we prove that the spatial simulation error of the finite difference method is associated with the dot product of the spatial dispersion relation of the finite-difference weights and the spectrum of the seismic wavefield. Based on the dot product relation, we construct a norm cost function to minimize spatial simulation error. For solving this optimization problem, the seismic wavefield information in wavenumber region is necessary. Nevertheless, the seismic wavefield is generally obtained by costly forward modeling techniques. To reduce the computational cost, we substitute the spectrum of the seismic wavelet for the spectrum of the seismic wavefield, as the seismic wavelet plays a key role in determining the seismic wavefield. In solving the optimization problem, we design an exhaustive search method to obtain the solution of the norm optimization problem. After solving the optimization problem, we are able to achieve the finite-difference weights that minimize spatial simulation error. In theoretical error analyses, the finite-difference weights from the proposed method can output more accurate simulation results compared to those from previous optimization algorithms. Furthermore, we validate our method through numerical tests with synthetic models, which encompass homogenous/inhomogeneous media as well as isotropic and anisotropic media.

Two-step optimization algorithm operated by heuristic and machine learning methods

Heuristics aim to provide approximate solutions for NP-hard combinatorial optimization (CO) problems in a reasonable time. They are faster than deterministic methods, where the solving process may take days. This paper presents an attempt to combine a heuristic algorithm and some machine learning (ML) methods for solving the Multidimensional Knapsack Problem (MKP), which is a well-known CO case. The approach named Genetic Algorithm and Machine Learning (GAML) is a two-step algorithm that first uses four machine methods to produce four solutions and then integrates the best one within the GA. For this purpose, (1) This approach builds models by training four machine learning (ML) methods using the decision variables obtained from the optimal solutions of simple MKP instances. The decision variable of an item is equal to one if it is selected in the knapsack and zero otherwise. (2) The four models are applied to predict the decision variables for complex MKP instances. (3) A dummy algorithm constructs feasible solutions from the predictions by adding the selected items randomly while the constraints are not violated. (4) For each MKP instance, the best feasible solution among the four built is integrated with the initialization and the crossing operators of a genetic algorithm. An experiment has been conducted on well-known complex MKP benchmarks. The results proved that the models were able to provide high-quality solutions. Also, the proposed algorithm was faster than other approaches in the literature.

METHODOLOGY’S OF THE SYSTEMIC APPROACH APPLICATION TO SOLVING TYPICAL TASKS OF PUBLIC ADMINISTRATION

The article is devoted to the study of the possibility of using a systemic approach in the processes of optimal resolution of typical problems of public administration entities. The results showed the possibility of using the specified methodology for the formation of a standard protocol for the optimal solution of typical problems of public authorities. In the work, based on the method of hierarchies, the typical problems of the activity of public authorities are singled out. A detailed correlational and functional analysis of the environment of the problems of public administration subjects and their causes was carried out. The existing applied scientific approaches to solving the problems of power activity were studied and the system approach was singled out as the main methodological tool for the complex solution of problems of this type. It has been proven that for an adequate understanding and modeling of the laws of development of complex systems and their management, it is necessary to use interdisciplinary research, which is based on a systemic approach and uses all modern scientific methods characteristic of both the humanities and exact sciences. Based on the existing methods of solving management problems, theoretical approaches to the formation of the sequence and content of solving typical problems of public authorities have been synthesized. A new vision of optimal approaches to solving problems of the specified class was formed based on an adaptive functional model of management activity. Based on a systemic approach, a meaningful algorithm for the modernization of the organizational and functional structure of public administration entities is proposed.

A digital design framework for the dimensional optimization of parallel robots based on kinematic and elasto-dynamic performance

The dimensional optimization based on kinematic and dynamic performance indices is proven significant for improving the operational performance of parallel robots. As the pose-dependent performances vary with the dimensional parameters, there are still many challenges in the optimal design of parallel robots, such as the possible conflict amongst different performance indices and computational expensiveness. By considering a 4-DOF high-speed parallel robot as an illustrative example, this paper presents a framework for the optimal design of parallel robots by integrating skeleton modeling, CAD-CAE integration and multi-objective optimization techniques. In this approach, the models for kinematic and elasto-dynamic performance evaluation are first developed using the CAD and CAE techniques, respectively. After evaluating the performance distributions over the task workspace, several reference poses representative of multiple neighboring sample poses are determined to formulate corresponding local performance indices by using the Hard C-Means (HCM) clustering analysis algorithm. This consideration leads to the determination of the optimal dimensions by investigating the Pareto-optimal solutions of a multi-objective optimization problem, allowing the conflict amongst different performance indices to be appropriately handled and the computational time to be saved significantly. The results of the case study show that the proposed approach can significantly enhance the elastic dynamic performance while ensuring that the kinematic performance is not excessively compromised, when compared to the performance of existing physical prototype of the robot. A software package has been developed using the proposed framework, providing a highly accessible way for designing various types of parallel/hybrid robots based on CAD and CAE techniques.

A Newton descent logarithmic barrier interior-point algorithm for monotone LCP

In this paper, based on the optimization techniques, we propose a descent logarithmic barrier interior-point method for solving monotone linear complementarity problems (LCP). The idea is to transform the LCP into a convex quadratic optimization problem, denoted (CQO). Then, the associated barrier problem to CQO is formulated. The existence and the uniqueness of optimal solution of the barrier problem is showed and its convergence to a solution of LCP is proved. For its numerical aspects, the descent direction is computed by using the classical Newton's method. However, to determine the displacement step along this direction, guaranteeing the maintenance of the new iterates inside the domain during the algorithm process and the improvement of the value of the objective function, we apply a new approach using approximation functions known as "minorant and majorant approximating functions". The numerical results obtained are very promising and show the effectiveness of this new strategy.

PIPO-Net: A Penalty-based Independent Parameters Optimization deep unfolding Network

Compressive sensing (CS) has been widely applied in signal and image processing fields. Traditional CS reconstruction algorithms have a complete theoretical foundation but suffer from the high computational complexity, while fashionable deep network-based methods can achieve high-accuracy reconstruction of CS but are short of interpretability. These facts motivate us to develop a deep unfolding network named the penalty-based independent parameters optimization network (PIPO-Net) to combine the merits of the above mentioned two kinds of CS methods. Each module of PIPO-Net can be viewed separately as an optimization problem with respective penalty function. The main characteristic of PIPO-Net is that, in each round of training, the learnable parameters in one module are updated independently from those of other modules. This makes the network more flexible to find the optimal solutions of the corresponding problems. Moreover, the mean-subtraction sampling and the high-frequency complementary blocks are developed to improve the performance of PIPO-Net. Experiments on reconstructing CS images demonstrate the effectiveness of the proposed PIPO-Net.

Approximate efficient solutions for nondifferentiable multiobjective optimization problems with an infinite number of constraints

Approximate efficient solutions for nondifferentiable multiobjective optimization problems with an infinite number of constraints

Solution existence for a class of nonsmooth robust optimization problems

AbstractThe main purpose of this paper is to investigate the existence of global optimal solutions for nonsmooth and nonconvex robust optimization problems. To do this, we first introduce a concept called extended tangency variety and show how a robust optimization problem can be transformed into a minimizing problem of the corresponding tangency variety. We utilize this concept together with a constraint qualification condition and the boundedness of the objective function to provide relationships among the concepts of robust properness, robust M-tamesness and robust Palais-Smale condition related to the considered problem. The obtained results are also employed to derive necessary and sufficient conditions for the existence of global optimal solutions to the underlying robust optimization problem.

Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality

In this paper, we present a Wolfe-type dual model containing the Caputo-Fabrizio fractional derivative, weak and strong duality results, number of Kuhn-Tucker type sufficient optimality conditions and duality results for variational problems (VPs) with Caputo-Fabrizio (CF) fractional derivative operator under weaker invexity assumptions. This newly developed fractional derivative operator delivers an exponential type kernel of nonsingular nature which characterizes the dynamics of physical systems and engineering processes with memory characteristics in a better way. This derivative operator is a convolution of first-order derivative and an exponential function. The proposed work also derives the global optimality criterion of the primal problem, Mond-Weir type duality results, and Mangasarian type strict converse duality theorem in view of this fractional differential operator possessing an exponential type kernel. The derived theorems investigate the global optimal solution of the primal problem. The main results of the present work are duality theorems and sufficient optimality conditions for VPs possessing the CF derivative. In view of applications of the derived optimality theorems, Mond-Weir type duality results have been established subjected to invexity assumptions. These applications and results generalize other important duality results of VPs and also provide results connected to duality with generalized invexity in mathematical programming. Several conventional results can also be seen as a special case of the obtained results in this work. 2010 Mathematics Subject Classification90C29; 90C46; 26A33

An Innovative Hybrid Approach Producing Trial Solutions for Global Optimization

Global optimization is critical in engineering, computer science, and various industrial applications as it aims to find optimal solutions for complex problems. The development of efficient algorithms has emerged from the need for optimization, with each algorithm offering specific advantages and disadvantages. An effective approach to solving complex problems is the hybrid method, which combines established global optimization algorithms. This paper presents a hybrid global optimization method, which produces trial solutions for an objective problem utilizing a genetic algorithm’s genetic operators and solutions obtained through a linear search process. Then, the generated solutions are used to form new test solutions, by applying differential evolution techniques. These operations are based on samples derived either from internal line searches or genetically modified samples in specific subsets of Euclidean space. Additionally, other relevant approaches are explored to enhance the method’s efficiency. The new method was applied on a wide series of benchmark problems from recent studies and comparison was made against other established methods of global optimization.

Privacy-preserving anomaly detection in stochastic dynamical systems: Synthesis of optimal Gaussian mechanisms

We present a framework for designing distorting mechanisms that allow the remote operation of anomaly detectors while preserving privacy. We consider a problem setting in which a remote station seeks to identify anomalies in dynamical systems using system input–output signals transmitted over communication networks. However, disclosing the true input–output signals of the system is not desired, as it can be used to infer private information. To maintain privacy, we propose a privacy-preserving mechanism that distorts input and measurement data before transmission using additive dependent Gaussian random processes and sends the distorted data to the remote station (which inevitably leads to degraded detection performance). We formulate constructive design conditions for the probability distributions of these additive processes while taking into account the trade-off between privacy, quantified using information-theoretic metrics (mutual information and differential entropy), and anomaly detection performance, characterized by the detector false alarm rate. The design of the privacy mechanisms is formulated as the solution of a convex optimization problem where we maximize privacy over a finite window of realizations while guaranteeing a bound on performance degradation of the anomaly detector.

Dynamic Supplier Selection and Its Optimal Strategy Considering Full Truck Load and Fuzzy Demand using Fuzzy Expected Value-Based Programming

This article proposes a linear integer optimization model incorporating fuzzy parameters to find the optimal solution for a dynamic supplier selection problem with uncertain demand. The uncertain demand value is represented using a fuzzy variable. A fuzzy expected value-based linear optimization solver is used to address the optimization problem, to minimize the total cost under fuzzy demand values. Several computational experiments were conducted to evaluate and analyze the model. The results show that the proposed model effectively identifies the optimal suppliers for each product. Additionally, the model determines the optimal purchase volumes for each product type from the selected suppliers, leading to the minimal total expected cost.