Process Knowledge-guided Autonomous Evolutionary Optimization for Constrained Multiobjective Problems

Various real-world problems can be attributed to constrained multi-objective optimization problems. Although there are various solution methods, it is still very challenging to automatically select efficient solving strategies for constrained multi-objective optimization problems. Given this, a process knowledge-guided constrained multi-objective autonomous evolutionary optimization method is proposed. Firstly, the effects of different solving strategies on population states are evaluated in the early evolutionary stage. Then, the mapping model of population states and solving strategies is established. Finally, the model recommends subsequent solving strategies based on the current population state. This method can be embedded into existing evolutionary algorithms, which can improve their performances to different degrees. The proposed method is applied to 41 benchmarks and 30 dispatch optimization problems of the integrated coal mine energy system. Experimental results verify the effectiveness and superiority of the proposed method in solving constrained multi-objective optimization problems.The National Key R&D Program of China, the National Natural Science Foundation of China, Shandong Provincial Natural Science Foundation, Fundamental Research Funds for the Central Universities and the Open Research Project of The Hubei Key Laboratory of Intelligent Geo-Information Processing.http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4235hj2023Electrical, Electronic and Computer Engineerin

On the Convergence of Newton-type Proximal Gradient Method for Multiobjective Optimization Problems

In a recent study, Ansary (Optim Methods Softw 38(3):570-590,2023) proposed a Newton-type proximal gradient method for nonlinear multiobjective optimization problems (NPGMO). However, the favorable convergence properties typically associated with Newton-type methods were not established for NPGMO in Ansary's work. In response to this gap, we develop a straightforward framework for analyzing the convergence behavior of the NPGMO. Specifically, under the assumption of strong convexity, we demonstrate that the NPGMO enjoys quadratic termination, superlinear convergence, and quadratic convergence for problems that are quadratic, twice continuously differentiable and twice Lipschitz continuously differentiable, respectively.Comment: arXiv admin note: text overlap with arXiv:2306.0979

Interactive optimization of mechanical systems with multiple performance requirements

In the design of mechanical systems, the designer is often faced with the problem of satisfying a number of competing requirements, including light weight and limits on displacements, stresses, and natural frequencies. This usually leads to an optimization problem with weight as the objective and the performance requirements as constraints. It may be desirable, in addition to minimizing weight, to minimize or maximize other performance measures rather than enforcing performance limits;This thesis recast the optimization problem as the minimization of multiple objectives, including performance indices and weight, subject to restrictions placed on the size of the design variables. The thesis also developed an interactive optimization procedure to enable design engineers to bring their skill into play during the optimization process;The procedure transforms the multi-objective problem into a single objective problem by taking the weighted sum of the performance indices and the structure\u27s weight. The weighting factors reflect the relative importance to the designer of the various conflicting objectives. The choice of the weighting factors best suited for the problem is not generally obvious and may require several adjustments before leading to an acceptable design. Thus the procedure is set up so that the designer can choose the weighting factors interactively;Optimization software was developed to provide designers with a decision making tool which is easy to use and provides useful information from which designer can confidently proceed. Since the method uses the NASTRAN for the response and sensitivity calculations, it is applicable to almost any structure which can be modelled using finite elements;Two examples illustrate the power of the technique. The first considers the redesign of an automotive engine block where the challenge was to find a low-weight design which has no natural frequencies in undesirable frequency bands. This problem illustrated the trade-offs between the weight and the natural frequencies, and it demonstrated the interactive process wherein the designer found a combination of the twelve design variables that met performance requirements with a very low weight structure. The second example concerned the redesign of the mounting structure of a heavy-duty truck\u27s exhaust pipe. The structural optimization was stated as finding a low weight structure which satisfied limits on maximum displacements and stresses, and shifted the natural frequencies out of an undesirable band. Again, the design engineer interactively found a very nice solution

Barzilai-Borwein Descent Methods for Multiobjective Optimization Problems with Variable Trade-off Metrics

The imbalances and conditioning of the objective functions influence the performance of first-order methods for multiobjective optimization problems (MOPs). The latter is related to the metric selected in the direction-finding subproblems. Unlike single-objective optimization problems, capturing the curvature of all objective functions with a single Hessian matrix is impossible. On the other hand, second-order methods for MOPs use different metrics for objectives in direction-finding subproblems, leading to a high per-iteration cost. To balance per-iteration cost and better curvature exploration, we propose a Barzilai-Borwein descent method with variable metrics (BBDMO\_VM). In the direction-finding subproblems, we employ a variable metric to explore the curvature of all objectives. Subsequently, Barzilai-Borwein's method relative to the variable metric is applied to tune objectives, which mitigates the effect of imbalances. We investigate the convergence behaviour of the BBDMO\_VM, confirming fast linear convergence for well-conditioned problems relative to the variable metric. In particular, we establish linear convergence for problems that involve some linear objectives. These convergence results emphasize the importance of metric selection, motivating us to approximate the trade-off of Hessian matrices to better capture the geometry of the problem. Comparative numerical results confirm the efficiency of the proposed method, even when applied to large-scale and ill-conditioned problems

Some Aspects of Mathematical Programming in Statictics

The Almighty has created the Universe and things present in it with an order and proper positions and the creation looks unique and perfect. No one can even think much better or imagine to optimize these further. People inspired by these optimum results started thinking about usage of optimization techniques for solving their real life problems. The concept of constraint optimization came into being after World War II and its use spread vastly in all fields. However, in this process, still lots of efforts are needed to uncover the mysteries and unanswered questions, one of the questions always remains live that whether there can be a single method that can solve all types of nonlinear programming problems like Simplex Method solves linear programming problems. In the present thesis, we have tried to proceed in this direction and provided some contributions towards this area. The present thesis has been divided into five chapters, chapter wise summary is given below: Chapter-1 is an introductory one and provides genesis of the Mathematical Programming Problems and its use in Statistics. Relationship of mathematical programming with other statistical measures are also reviewed. Definitions and other pre-requisites are also presented in this chapter. The relevant literature on the topic has been surveyed. Chapter-2 deals with the two dimensional non-linear programming problems. We develop a method that can solve approximately all type of two dimensional nonlinear programming problems of certain class. The method has been illustrated with numerical examples. Chapter-3 is devoted to the study of n-dimensional non-linear programming problems of certain types. We provide a new method based on regression analysis and statistical distributions. The method can solve n-dimensional non-linear programming problems making use of regression analysis/co-efficient of determination. In chapter-4 we introduce a filtration method of mathematical programming. This method divides the constraints into active and non active and try to eliminate the less important constraints (non-active constraints) and solve the problem with only active constraints. This helps to find solution in less iterations and less in time while retaining optimality of the solution. The final chapter-5 deals with an interesting relationship between linear and nonlinear programming problems. Using this relationship, we can solve linear programming problems with the help of non-linear programming problems. This relationship also helps to find a better alternate solutions to the linear programming problems. In the end, a complete bibliography is provided

Reservoir management under consideration of stratification and hydraulic phenomena

Reservoirs are the most important components in a water resources system. They are used to store water to extend its temporal availability. The physical, chemical and biological characteristics of water change when impounded in reservoirs. This implies the possibility of using reservoirs for the control of the quality of water besides merely satisfying the quantity requirement. This study presents several techniques formulated to manage a reservoir when both quantity and quality of water are of interest. In this study salinity is selected to characterize the water quality status. The approaches are demonstrated using data from the Jarreh Reservoir on the Shapur river in Iran.Water in a reservoir is stratified for most of a year due to difference in density caused by temperature, dissolved and suspended solids. Therefore, in a stratified reservoir the quality of water that is interrelated to density varies with depth. Consequently, this feature could be used in the process of reservoir operational policy determination to improve the quality of water supply. The aim of this research is to analyze different approaches regarding the incorporation of this phenomenon into reservoir operational policies and to propose those which require the least increase in mathematical and computational complexity.Initially, two techniques that rely on the natural process of stratification occurring in a reservoir are presented. The first methodology proceeds stepwise in time alternating optimization and simulation of reservoir operation at each time step. A one-dimensional reservoir dynamics simulation model is employed to simulate the stratification of the reservoir. A constrained nonlinear optimization model is used to identify optimum releases. In the optimization step the reservoir is assumed to be equivalent to the parallel configuration of several smaller hypothetical reservoirs, the number of which being equal to the number of outlets. There is no communication among these hypothetical reservoirs. The applicability of the technique is tested for three hydrologically different years and for a continuous period of five years. Incorporation of inflow stochasticity into the methodology is devised through the integration of an optimization model based on Stochastic Dynamic Programming technique.Next, an iterative technique, in which an optimization model and a reservoir stratification simulation model operate interactively, is presented. One iteration cycle comprises the run of the optimization model and the simulation model: i) Reservoir operation is optimized over the entire time period (year); ii) Simulation of stratification is applied over the entire time period. The optimization model is based on Incremental Dynamic Programming technique. In the optimization model, the hypothetical reservoir concept used in the above model is adopted. However, communication between any two adjoining hypothetical reservoirs is allowed in the model. The one-dimensional reservoir dynamics simulation model simulates the stratification of the reservoir. The applicability of the technique is examined for three hydrologically different years.Reservoirs could also be modelled by assuming that complete mixing of water is occurring throughout its entire volume during a year. It is a simplification as compared with the real behaviour of stratification occurring in reservoirs. Two models are developed based on this assumption to improve the quality of water supply. In one model only the releases are controlled. In the other, both inflows and releases are controlled. Optimization is based on Incremental Dynamic Programming technique. The results from both models show improvements in the quality of water supplied from the reservoir. However, the improvements obtained by manipulating both inflows and releases are more profound.Improving the quality of water supplied from a reservoir by diverting poor quality inflows and satisfying downstream quantity demands are two conflicting objectives. This problem is studied under the multiobjective analysis framework. The reservoir is assumed to be completely mixed throughout its volume during the whole annual cycle. The results show that a cautious balance between the quantity of water supplied for downstream and the volume of inflows diverted would lead to marked reduction in the supply salinity.The study reveals that the quality of reservoir releases could be improved by withdrawals from different elevations in a stratified reservoir. However, the benefits obtained in this way are marginal for the case study reservoir. Similar improvements are observed under the assumption that the reservoir is completely mixed throughout a year. On the other hand, by manipulating the inflows to the Jarreh reservoir these improvements could be enhanced significantly. That is, by-passing of poor quality inflows seems to be a very promising management alternative for improving the quality of water supplied from the reservoir. The assumption of reservoir's complete mixing is warranted for the stratified reservoir by the obtained results. Hence, a relatively simple and straightforward methodology based on the non-stratification assumption proves to be suitable in managing a density stratified reservoir