Solution Of Optimization Problem Research Articles

HPB3C-3PG algorithm: A new hybrid global optimization algorithm and its application to plant classification

This paper proposes a hybrid bio-inspired search and optimization algorithm that combines the strengths of the PB3C (Parallel Big Bang Big Crunch) and 3PGA (3 Parent Genetic Algorithm) algorithms. The hybrid algorithm employs a single population-based evolutionary search coupled with multi-population parallel processing techniques to address optimization problems. The proposed algorithm is implemented in MATLAB software. We evaluate the performance of the proposed algorithm on the CEC2021 standard test bench suite. The performance of the proposed approach is compared with that of the other nine algorithms. The comparative analysis shows that the proposed hybrid PB3C and 3PGA algorithms performed better than the other nine optimization algorithms. Furthermore, this chapter proposes an HPB3C-3PGA-based approach to evolve the near-optimal architecture of CNN. The proposed plant image classification approach is implemented in Python and compared with 12 other approaches. The proposed approach achieved an accuracy of 98.96% on the Mendeley dataset and 98.97% on the CVIP100 dataset. The proposed approach outperforms all other approaches for the plant leaf classification problem. This research significantly contributes to overcoming limitations in existing approaches, providing a robust solution for optimization problems and image classification tasks.

Pricing European call options with interval-valued volatility and interest rate

We propose a novel approach to pricing European call options when both of the volatility of the underlying asset and interest are uncertain. In this approach, we formulate the option pricing problem with uncertain parameters as a partial-differential inequality constrained interval optimization problem. An interior penalty method is then developed for the numerical solution of the finite-dimensional optimization problem arising from the discretization of the continuous pricing problem by a finite difference scheme. A convergence theory for the penalty method is established. An algorithm based on Newton's iterative method is also proposed for solving the penalty equation. Numerical results are presented to demonstrate the effectiveness and usefulness of this approach and the numerical methods.

Optimal control for a nonlinear Schrödinger problem perturbed by multiplicative fractional noise

The aim of this paper is to investigate the existence of optimal solutions for control problems involving nonlinear stochastic Schrödinger equations perturbed by a multiplicative fractional Brownian motion. Finite-dimensional approximations and a linearization method are employed and lead to ε-optimal solutions for the considered problems.

Learning to Build Solutions in Stochastic Matching Problems Using Flows (Student Abstract)

Generative Flow Networks, known as GFlowNets, have been introduced in recent times, presenting an exciting possibility for neural networks to model distributions across various data structures. In this paper, we broaden their applicability to encompass scenarios where the data structures are optimal solutions of a combinatorial problem. Concretely, we propose the use of GFlowNets to learn the distribution of optimal solutions for kidney exchange problems (KEPs), a generalized form of matching problems involving cycles.

Inexact direct-search methods for bilevel optimization problems

In this work, we introduce new direct-search schemes for the solution of bilevel optimization (BO) problems. Our methods rely on a fixed accuracy blackbox oracle for the lower-level problem, and deal both with smooth and potentially nonsmooth true objectives. We thus analyze for the first time in the literature direct-search schemes in these settings, giving convergence guarantees to approximate stationary points, as well as complexity bounds in the smooth case. We also propose the first adaptation of mesh adaptive direct-search schemes for BO. Some preliminary numerical results on a standard set of bilevel optimization problems show the effectiveness of our new approaches.

Bi-Basis Function Second Derivative Block Method for the Solution of Optimal Control Problems

This paper determines the state trajectory of a dynamic system over a period to optimize a given performance index. A one-step second derivative hybrid method based on the combination of Lucas and first Boubaker polynomials as basis functions is formulated for the solution of optimal control problems via the forward-backward sweep method. The technique is used to generate a set of hybrid schemes at selected grid and off-grid points with implementation in block form. The stability and convergence of the new method is discussed and the accuracy of the method is tested on some numerical examples. The results obtained correspond to a large extent with the results obtained by the exact solution. Bi-bases function, Block method, First Boubaker polynomials, Lucas polynomial, Optimal control problem

Parameterized complexity for iterated type partitions and modular-width

This paper deals with the complexity of some natural graph problems parameterized by some measures that are restrictions of clique-width, such as modular-width and neighborhood diversity. We introduce a novel parameter, called iterated type partition number, that can be computed in linear time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parameterized by the iterated type partition number. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parameterized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity.Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition number, which enables us to find optimal solutions for several graph problems. As an example, in this paper, we present algorithms parameterized by the iterated type partition number of the input graph for some generalized versions of the Maximum Clique, Minimum Graph Coloring, (Total) Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems. Each algorithm outputs not only the optimal value but also the optimal solution. We stress that while the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. We finally show that the proposed scheme can be used to devise polynomial kernels, with respect to iterated type partition number, for the decisional version of most of the problems mentioned above.

Separable programming problem with bipolar max-T fuzzy relation equation constraints

One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max-T composition operator, where T is a continuous and Archimedean t-norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max-continuous and Archimedean t-norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.

An asymptotically optimal solution for the minimization of the variation of job completion times in two-stage proportionate no-wait flow shops

We propose an asymptotically optimal solution for the two-stage no-wait proportionate flow shop scheduling problem with the objective of minimizing the total absolute deviation of job completion times. The proposed solution is within 1% of the optimum for problems with more than 20 jobs. We also show that the problem is solved optimally in constant time for the special case where the two smallest jobs have equal processing times. Our results provide answers to most of the outstanding research questions for this problem.

Distributed Real-Time Feedback Optimization for Renewable Energy Sources and Vehicle-to-Grid Power Compensation of Electric Vehicle Chargers in Distribution Systems

A novel distributed feedback optimization-based controller for electric vehicle (EV) chargers and renewable energy sources (RESs) in distribution systems is proposed. The proposed controller utilizes the flexibility in EV chargers’ active and reactive power consumption to offer the desirable vehicle-to-grid services. Instead of using the conventional cascaded PI controllers, a new optimization-based approach is proposed to control RESs to track their power injection setpoints. The proposed controller formulates the control targets as a single constrained optimization problem, i.e., to minimize the critical bus voltage magnitude deviations while driving RESs to follow their power setpoints, thereby fulfilling the EV charging requirements and regulating their power outputs and bus voltage magnitudes to stay within their limits. A distributed feedback optimization-based control algorithm is designed for EV chargers and RESs to steer the system trajectories of the distribution systems towards the optimal solution of the formulated optimization problem. Simulation results show that the proposed controller can always steer the test system to the optimal solution of the optimization problem. The advantages of the real-time vehicle-to-grid power compensation of EV chargers are also demonstrated.

Optimal Design of Sparse Matrix Phased Array Using Simulated Annealing for Volumetric Ultrasonic Imaging with Total Focusing Method.

The total focusing method (TFM) is often considered to be the 'gold standard' for ultrasonic imaging in the field of nondestructive testing. The use of matrix phased arrays as probes allows for high-resolution volumetric TFM imaging. Conventional TFM imaging involves the use of full matrix capture (FMC) for ultrasonic signals acquisition, but in the case of a matrix phased array, this approach is associated with a huge volume of data to be acquired and processed. This severely limits the frame rate of volumetric imaging with 2D probes and necessitates the use of high-end equipment. Thus, the aim of this research was to develop a novel design method for determining the optimal sparse 2D probe configuration for specific conditions of ultrasonic imaging. The developed approach is based on simulated annealing and involves implementing the solution of the sparse matrix phased array layout optimization problem. In order to implement simulated annealing for the aforementioned task, its parameters were set, the acceptance function was introduced, and the approaches were proposed to compute beam directivity diagrams of sparse matrix phased arrays in TFM imaging. Experimental studies have shown that the proposed approach provides high-quality volumetric imaging with a decrease in data volume of up to 84% compared to that obtained using the FMC data acquisition method.

Efficient solutions for vector optimization problem on an extended interval vector space and its application to portfolio optimization

In this paper, a generalized interval vector space is investigated and defined as an ordered relation in the form of a bijective linear transformation of its onto a real vector space. The ordered relation is utilized to formulate an interval optimization problem in the same manner as a classical multi-objective programming problem. A methodology that addresses the existence of efficient solutions for the multi-objective interval optimization problem has also been discussed. Various numerical examples are described to illustrate and substantiate all developed concepts. Furthermore, the multi-objective portfolio rebalancing problem for a time horizon is designed based on the developed interval vector optimization. An algorithm using exhaustive solution technology has been proposed to achieve an efficient investment strategy. Finally, its applicability and efficacy are analyzed using Bombay Stock Exchange India data sets.

An accelerated Benders decomposition approach for virtual base station formation in stochastic Cloud-RANs

In Cloud-RANs, the baseband unit (BBU) is separated from the remote radio head (RRH) at the cell site and migrated to a server cloud to decrease deployment and maintenance costs. A base station (BS) can be formed as a network slice in which the virtual network functions (VNFs) of its BBU can be placed on any server by network virtualization (NFV) technology and logically connected by software-defined networking (SDN). However, the uncertain demands of users make it very difficult to determine a network slice with appropriate VNF placement and flow routing between VNFs. We aim to minimize the cost of providing BS services while meeting their time-varying demands. We consider the network slice scheme and fronthaul establishment jointly and formulate this as an integer linear programming (ILP) problem. We propose scenario classification Benders decomposition (CBD) based on Benders decomposition (BD) but with fewer Benders cuts, accelerating the solution convergence. Moreover, we propose an efficient heuristic approach to obtain good solutions for the ILP problem based on the solutions of its relaxed problems obtained by BD or CBD. The numerical experiments show that when combining our heuristic approach with CBD, the results are close to the optimal solutions of the ILP problem, while the computation time is significantly shorter than those of previous approaches.

A direct transcription method for solving distributed-order fractional optimal control problems

This paper considers the numerical solution of distributed-order Fractional Optimal Control Problems (FOCPs). The common approaches for solving these problems involve approximating the problem with a multi-term FOCP and subsequently discretizing the resulting multi-term problem using either a direct or indirect method. However, this paper bypasses the need for approximation with a multi-term FOCP by introducing innovative approximation formulas for distributed-order fractional derivatives. These approximation formulas are directly derived from the conventional Grünwald–Letnikov, L1, and trapezoidal formulas. Building upon these formulas, we develop the relevant fractional-order distributed-order operational matrices. Utilizing the operational matrix, we easily discretize the distributed-order FOCP to a Non-Linear Programming (NLP), which can be solved by a suitable NLP-solver. Additionally, to enhance the efficiency of solving the resulting NLP, we determine the closed-forms of both the Jacobian of constraints and the gradient of the objective function. The presented method is characterized by its speed, simplicity, and ease of implementation. Moreover, it can be used to solve a wide range of distributed-order FOCPs, such as those involving nonlinear dynamics, free final times, free terminal conditions, and path constraints. By means of ample numerical tests, the accuracy and efficiency of the proposed method are assessed.

Pointwise Error Estimates of Numerical Solutions to Linear Quadratic Optimal Control Problems

An auxiliary optimal control problem is formulated that provides with its unique solution, a continuous representation of the global error of a numerical approximation to the solution of a linear quadratic optimal control problem. The resulting error functions are characterized as the unique solutions to an optimality system that is reformulated as a boundary value problem. With this formulation, reliable pointwise error estimates can be generated utilizing well-established techniques of defect control. A novel algorithm based on defect correction and defect control is presented that generates pointwise approximations to the global error of numerical optimal control solutions on a uniform grid. It is proven and numerically validated that this algorithm can generate pointwise estimates that approximate the true global error with a prescribed accuracy.

User scheduling for non-orthogonal multiple access-enabled 5G aeronautical mobile airport communications system

The recent concept of aeronautical mobile airport communications system with the 5th generation mobile communication technology (5G AeroMACS) for enhancing real-time and efficient flight information transmission has attracted wide attention from the civil aviation industry. In addition, non-orthogonal multiple access (NOMA) is also a promising solution to further improve the sum rate. However, the fundamental limit of the conventional 5G AeroMACS is that, the base station (BS) equipped with dozens of antennas cannot support more users than its number of antennas using the same time-frequency resources. To break this limit, for the first time, we investigate user scheduling policies for the integration of NOMA in 5G AeroMACS, i.e., 5G AeroMACS-NOMA, employing a half-duplex (HD) BS for serving multiple HD downlink users simultaneously. The two proposed user scheduling policies are obtained from the sub-optimal solution of mixed combinatorial non-convex optimization problems for the maximization of the achievable sum rate of the entire system. To mitigate the inter-beam interference (IBI), a user clustering algorithm based on the principle of maximizing the channel spatial correlation is designed, and an iterative power allocation algorithm is utilized for maximizing the achievable sum rate of the entire system. Furthermore, to mitigate the serious error propagation caused by the nature in NOMA, a scenario with user division is considered, and an user scheduling policy based on successive convex approximation is designed to strike a balance between computational complexity and optimality. Simulation results show that the proposed 5G AeroMACS-NOMA schemes can achieve a higher achievable sum rate compared with conventional 5G AeroMACS.

A Convergent Legendre Spectral Collocation Method for the Variable-Order Fractional-Functional Optimal Control Problems

In this paper, a numerical method is applied to approximate the solution of variable-order fractional-functional optimal control problems. The variable-order fractional derivative is described in the type III Caputo sense. The technique of approximating the optimal solution of the problem using Lagrange interpolating polynomials is employed by utilizing the shifted Legendre–Gauss–Lobatto collocation points. To obtain the coefficients of these interpolating polynomials, the problem is transformed into a nonlinear programming problem. The proposed method offers a significant advantage in that it does not require the approximation of singular integral. In addition, the matrix differentiation is calculated accurately and efficiently, overcoming the difficulties posed by variable-order fractional derivatives. The convergence of the proposed method is investigated, and to validate the effectiveness of our proposed method, some examples are presented. We achieved an excellent agreement between numerical and exact solutions for different variable orders, indicating our method’s good performance.

Optimizing defect rework for a software start-up's schedule variation: a Six-Sigma-based approach

PurposeManaging project completion within the stipulated time is significant to all firms' sustainability. Especially for software start-up firms, it is of utmost importance. For any schedule variation, these firms must spend 25 to 40 percent of the development cost reworking quality defects. Significantly, the existing literature does not support defect rework opportunities under quality aspects among Indian IT start-ups. The present study aims to fill this niche by proposing a unique mathematical model of the defect rework aligned with the Six Sigma quality approach.Design/methodology/approachAn optimization model was formulated, comprising the two objectives: rework “time” and rework “cost.” A case study was developed in relevance, and for the model solution, we used MATLAB and an elitist, Nondominated Sorting Genetic Algorithm (NSGA-II).FindingsThe output of the proposed approach reduced the “time” by 31 percent at a minimum “cost”. The derived “Pareto Optimal” front can be used to estimate the “cost” for a pre-determined rework “time” and vice versa, thus adding value to the existing literature.Research limitations/implicationsThis work has deployed a decision tree for defect prediction, but it is often criticized for overfitting. This is one of the limitations of this paper. Apart from this, comparing the predicted defect count with other prediction models hasn’t been attempted. NSGA-II has been applied to solve the optimization problem; however, the optimal results obtained have yet to be compared with other algorithms. Further study is envisaged.Practical implicationsThe Pareto front provides an effective visual aid for managers to compare multiple strategies to decide the best possible rework “cost” and “time” for their projects. It is beneficial for cost-sensitive start-ups to estimate the rework “cost” and “time” to negotiate with their customers effectively.Originality/valueThis paper proposes a novel quality management framework under the Six Sigma approach, which integrates optimization of critical metrics. As part of this study, a unique mathematical model of the software defect rework process was developed (combined with the proposed framework) to obtain the optimal solution for the perennial problem of schedule slippage in the rework process of software development.

Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions

In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example. Funding: This work was supported by OP RDE [Grant CZ.02.1.01/0.0/0.0/16_019/0000765].

Optimal design of truss domes with frequency constraints using seven metaheuristic algorithms incorporating a comprehensive statistical assessment

In this article, seven recently developed metaheuristic algorithms are utilized for optimal design of dome-shaped trusses with natural frequency constraints. These algorithms are social network search, jellyfish search optimizer, equilibrium optimizer, teaching-learning-based optimization, grey wolf optimizer, colliding bodies optimization, and the improved grey wolf optimizer. This study focuses on the assessment and tuning of these algorithms to enhance both computational efficiency and solution quality. In particular, we introduce a hyperparameter tuning procedure that significantly improves the performance of the algorithms to make them more suitable for the solution of structural optimization problems. The performance of the selected algorithms is studied through five benchmark examples for design optimization of dome structures, from small-scale to large-scale cases, with frequency constraints. Many combination scenarios are studied for hyperparameter tuning of the algorithms in each design example in order to provide comprehensive statistical results determining the sufficient number of structural analyses as well as the number of particles to find an efficient condition of each algorithm. Indeed, this study also presents an insight into the efficient setting of hyperparameters for each algorithm, leading to a better solution with lower computational cost. Furthermore, the optimization results provide a comparison of the solutions obtained in the current and previous studies in order to demonstrate the suitability of the selected algorithms.