Nonparametric methods constitute a highly significant > To address this issue, in 2016, Morovati et al. proposed a generalization of Barzilai-Borwein methods for multi-objective optimization problems. Compared to other gradient-based methods, their approach demonstrated notably higher accuracy and speed, which drew significant interest from researchers in this area. More recently, another generalization of the Barzilai-Borwein method has been introduced, in which a specific weight is assigned to each objective function. The aim of assigning these weights is to reduce the adverse effect of conflicts among objectives on the step size reduction. The authors of the respective study compared their proposed method with that of Morovati et al., and their analysis and numerical results state that their proposed approach has substantial superiority. However, in this thesis, this issue is investigated more critically and thoroughly. It is demonstrated that the above-mentioned analysis is based on a form of false convergence for the newly proposed method and an improper comparison between the two approaches. In fact, it is shown that, contrary to the claims in the literature, the method of Morovati et al. has significant advantages over the proposed approach. This superiority is confirmed both on the test problems presented in the prior study and on a much broader set of ‎benchmark‎  

Iterative methods for solving multi-objective optimization problems have greater computational complexity compared to single-objective problems. Accordingly, gradient-based methods that do not use higher-order derivatives are more desirable for this purpose. On the other hand, these methods have slower convergence rates. One important idea to address this issue is to use information from previous iterations alongside the gradient of the current iteration to construct the desired direction. The most basic methods based on this idea are conjugate gradient methods. In this regard, this thesis addresses some methods that, by employing suitable parameters and utilizing information from previous iterations, yield relatively fast processes for solving multi-objective optimization problems. While investigating the convergence of these methods, their computational superiority is demonstrated using some standard test problems in multi-objective optimization.  

  ‎Because ‎of ‎the ‎shortcomings ‎and ‎numerous ‎challenges ‎that scalarization methods face in solving multiobjective optimization problems, there has been a great deal of interest in recent years in the use of nonparametric methods, which are a generalization of iterative methods in singleobjective optimization.‎‎However, less attention has been paid to the study of accelerated versions of these algorithms. in this paper, an accelerated proximal gradient algorithm is studied for solving multiobjective optimization algorithms in which each objective function is the sum of a differentiable convex function and a proper convex function.‎‎This method, also known as the Fast Iterative Shrinkage- Thresholding Algorithm(FISTA), for scalar optimization. ‎‎The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case, which is ineffective in scalar ‎optimization.‎‎Furthermore, an efficient way to solve the subproblem via its dual representation is presented, and the validity of the proposed method is demonstrated through some numerical experiments

در اين پايان نامه مروري بر انواع روش هاي تقريب براي تجزيه مقاديرتكين[1] براي يك ماتريس را خواهيم داشت.تجزيه SVD ،از اساسي ترين تجزيه هاي جبر خطي عددي به شمار مي رود كه روي ماتريس هاي مستطيلي m*n بكار گرفته مي شود كه امروزه در دنياي پردازش تصوير در زمينه ي فشرده سازي كاربرد فراواني دارد.تجزيه مقدار تكين منجر مي شود تا تصوير را با ابعادي كوچكتر ذخيره كنيم به طوريكه اطلاعات اصلي آن همچنان حفظ شوند.براي فشرده سازي تصوير با استفاده از تجزيه SVD از تقريب ماتريس استفاده مي كنيم كه در فصل دوم اين پايان نامه تقريب و فشرده سازي ماتريس را شرح داده ايم.تقريب ماتريس منجر مي شود تا تصوير را با يك مجموعه ي كوچكتر و در عين حال اصلي تر از مقادير تكين مشاهده كنيم كه روش هاي مختلفي براي اين تقريب مانند نمودار Scree،قانون گاتمن_كايزر،آستانه سخت مقادير تكين و روش هاي مبتني بر آنتروپي،تحليل و ارزيابي مي شوند. در پايان نشان مي دهيم كه هركدام از اين روش ها در شرايط خاصي عملكرد بهينه اي دارند و بسته به ويژگي هاي داده ها،انتخاب مناسب يك روش، مي تواند بهينه ترين مقادير منفرد را براي تحليل و پردازش حفظ كند.    [1] Singular Value Decomposition      

يك خانواده مهم از اين اصلاحات روش هاي گراديان با تآخيرمي باشد. روش هاي گراديان با تاخير (GMR) يك روش تكراري غير يكنوا است كه براي حل معادلات خزي بزرگ متقارن و همچنين معين مثبت توسعه يافته است. اين روش تعميمي از روش هاي تندترين شيب و برزيليا-بوروين است.در اين پايان نامه همگرايي R-خطي اين روش اثبات مي شود و همچنين برخي از روش هاي گراديان دوري مورد مطالعه قرار مي گيرد.بررسي طول گام يوان و خواص طيفي روش هاي گراديان از ديگراهداف اين پايان نامه مي باشد.

Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor   networks. Inthiset we study the notion of excess for woven frames and prove that any two frames in a separable Hilbert space that are woven have the same excess. We also show that every frame with a large class of duals is woven provided that its redundant elements have small enough norm. Also, we try to transfer the woven property from frames to their duals and vice versa. Finally, we look at which perturbations of dual frames preserve the woven property

dimension frames.Keyword: Frame, Tight frame, Parseval frame, synthesis operator , Analysis operator, Frame operator, Dual frame, Span, Redundancy, Upper redundancy, Lower Redundancy, orthonormal basis

   Abstract This thesis introduces new concepts of quasi efficiency and quasi proper efficiency for multiobjective optimization problems. These concepts reduce to the most important existing concepts of approximate and quasi efficient solutions. Through the use of quasi efficient solutions, a generalized subdifferential of a vector mapping is introduced, which unifies a number of approximate subdifferentials frequently used in optimization. The general subdifferential is connected to the traditional subdifferential of real functions through scalarization. The use of a generalized subdifferential is employed to express optimality conditions for quasi-efficient solutions. Additionally, this thesis provides optimality conditions for multiobjective optimization problems with cone constraints and polyhedral ordering cones, focusing on approximate proper solutions. A first >Key words: Multiobjective optimization, Quasi e?ciency, approximate solutions, Linear scalarization, Nonlinear scalarization, Vector subdi?erential, Coradiant set, Optimality conditions.   

‎Gradient‎ family methods are known as one of the most important methods for solving unconstrained optimization problem. ‎S‎pectral gradient methods are an extensions o‏f gradient methods which aims to overcome some of the drawbacks of gradient methods and produce some efficient methods.‎ ‎ ‎ ‎In this thesis‎, the first goal is to study a new family of spectral gradient methods ‎while ‎t‎his family uses a new stepsize which is determined by a convex combination of the long and short Barzilai–Borwein (BB) stepsizes.‎‎‎ It is also shown that each member of this family have some appropriate quasi-Newton properties.‎‎ ‎ In the sequel, the convergence properties of the new algorithm is investigated and it is shown that the new family of methods is R-superlinearly convergent for two-dimensional problems and R-linearly convergent for the any-dimensional cases. ‎‎‎‎‎‎ In the second part of this thesis, ‎some ‎of ‎cyclic ‎gradient ‎methods ‎have ‎been ‎studied ‎and a‎ ‎new cyclic ‎gradient ‎method ‎and ‎its ‎conve‎rgence properties is studied.‎  

  AbstractOne of the important concepts from the point of view of theory and computation is the con-cept of proper efficiency in multi-objective optimization. On the other hand, in computationalprocesses, we usually obtain approximate solutions. Therefore, it is necessary to study the prop-erties of these types of solutions and approximate solutions to the related scalar problems to beexamined. Based on this, in this thesis, a generalization of the concept of proper efficiency forproblems with an infinite number of objective functions is investigated. It turns out that someresults for ordinary multi-objective problems cannot be generalized to these problems. In addition,some scalarization methods such as weighted sum and the Chebyshev method are presented re-lated to properly efficient solutions to these problems. In addition, a unified method based on thedirectional Pascoletti–Serafini approach is presented to find efficient, properly efficient, and weaklyefficient solutions as well as similar approximate solutions. In the analysis of these solutions, whilepresenting some characterizations, simple and implementable optimality conditions for efficient

Network Functions Virtualization of architecture means providing various network services without the need for hardware and not depending on it. Network Functions Virtualization is a new field in the network, with the help of which hardware devices can be implemented in virtual and software form. Network Functions Virtualization improves network functions such as: proxies, firewalls, load balancing, etc. In other words, using virtualization technology, this architecture is able to convert hardware devices into software modules known as virtual network functions and provide the desired service to the user. Providing the service requested by the user in the network is done by a sequence of virtual network functions, which are known as service functions chain. One of the main challenges in the development of network functions virtualization architecture is the allocation of resources to the requested network services in network infrastructures based on network functions virtualization, this challenge is called network function virtualization resource allocation problem. Therefore, in this research, the problem of allocating resources to virtual network functions in Network Functions Virtualization architecture has been solved by using mathematical programming techniques. In this research, a multi-objective mixed integer linear programming model is presented for the problem of resource allocation to virtual network functions. In this model, constraints related to the resource capacity of nodes and connections and delay constraints are desired. Also, the objective functions in this research are: maximum flows accepted in the network, reduction of resource costs of nodes (including: the number of CPU cores and the amount of memory), reduction capital costs, reduction operational costs and checking execution time. These constraints and objective functions are expressed precisely and explicitly by mathematical functions. The proposed mathematical model is implemented and solved with the Cplex solver. To evaluate the proposed mathematical model, several different topology are considered. The optimal cost is evaluated under changing parameters such as the length of service functions chain, the number of flows, the length of flows, the amount of resources of nodes, the number of nodes and the number of virtual network functions. And finally, the increase in execution time is checked by changing the number of nodes and the number of virtual network functions. The numerical results of this research show the effectiveness of the model in resources allocation to virtual network functions. Keywords: Network Functions Virtualization architecture, Virtualized Network Functions, Resource allocation, Mathematical programming, Mixed integer linear programming   

  In recent years, finding the closest target for the under-evaluation decision making units (DMU) has attracted the attention of researchers significantly, and numerous articles have been published in this field. In some of these articles, the related efficiency measure does not satisfy the strong monotonicity property. Since this property plays a very important role in comparing and ranking units, it is very desirable to present methods that, while finding the closest efficient model, their efficiency measure is strongly monotone. Mainly, researches done in this field are divided into the following two general categories: A) The methods that obtain all of the full dimensional efficient facets or their extended versions and then obtain the distance of the DMU under evaluation to these facets. B) Methods that, instead of obtaining full dimensional efficient facets, using some mixed integer linear programming models, implicitly calculate the distance of the under evaluation DMU to the strongly efficient frontier. In both cases, based on the obtained distance, a well-defined strong efficiency measure is introduced. This thesis investigates these methods in detail using some real numerical results.

  AbstractIn this thesis, two methods for solving the system of nonlinear equations with largedimensions are investigated. The first method is a hybrid conjugate gradient methodbased on the convex combination of Fletcher–Reeves (FR) and Polak–Ribière–Polyak(PRP) parameters. The global convergence of this algorithm is discussed. Numericalresults show the efficiency and accuracy of the method for problems with largedimensions. In the second method, a spectral conjugate gradient method based on theprojection method has been used for systems of nonlinear monotone equations. Also,this method is suitable for solving with large dimensions problemsKeywords:Conjugate gradient method, Convex combination, Self adaptive, Spectral conjugategradient method, Nonlinear monotone equations.

    ABSTRACT Two transformations of gradient-descent iterative methods for solving unconstrained optimization are proposed. The first transformation is called modification and it is defined using a small enlargement of the step size in various gradient-descent methods. The second transformation is termed as hybridization and it is defined as a composition of gradient-descent methods with the Picard–Mann hybrid iterative process. As a result, several accelerated gradient-descent methods for solving unconstrained optimization problems are presented, investigated theoretically and numerically compared. The proposed methods are globally convergent for uniformly convex functions satisfying certain condition under the assumption that the step size is determined by the backtracking line search. In addition, the convergence on strictly convex quadratic functions is discussed. Numerical comparisons show better behaviour of the proposed methods with respect to some existing methods in view of the Dolan and Moré’s performance profile with respect to all analysed characteristics: number of iterations, the CPU time, and the number of function evaluations.  KEYWORD : Unconstrained Optimization; Gradient-Descent methods; Muiti Step-Size; Convergence; line Search.

   دستگاه معادلات غيرخطي يكي از مسائل مهم و پركاربرد در رياضيات است. روش‌هاي متفاوتي براي حل اين مسائل تاكنون ارائه شده است. از ميان روش‌هاي تكراري براي حل اين مسائل، مي‌توان به روش نيوتون، روش‌هاي شبه نيوتن و نسخه‌هاي تغيير يافته آن‌ها اشاره كرد.يكي از نقاط ضعف مهم اين روش‌ها بخصوص براي مسائل با ابعاد بزرگ، نياز به محاسبه ماتريس ژاكوبي در هر تكرار و حل دستگاه معادلات خطي متناطر است. تلاش براي ارائه روش‌هاي بدون ژاكوبي براي حل دستگاه‌هاي معادلات غيرخطي در سال‌هاي اخير همواره مورد توجه محققان بوده است. در حالات خاص كه دستگاه معادلات داراي خواص ويژه مي‌باشد، الگوريتم‌هاي بسيار موثري معرفي شده‌اند. يكي از اين رده‌هاي خاص، دستگاه معادلات غيرخطي يكنوا مي‌باشد كه روش‌هاي حل متفاوتي براي آن ارائه شده است. يكي از مهمترين رده هاي موجود براي حل اين مسائل، الگوريتم‌هاي   مبتني بر تصوير است كه بواسطه نياز به حافظه كم، در حل دستگاه معادلات غيرخطي مقياس بزرگ يكنوا كاربردهاي زيادي دارند.   هدف اين پايان‌نامه، ارائه دو خانواده جديد از الگوريتم‌هاي بدون مشتق مبتني بر تصوير است كه از جهاتي شبيه جهات گراديان مزدوج سه‌جمله‌ا‌ي استفاده مي كنند جاييكه ثابت مي شود جهات تعريف شده در شرايط كاهش كافي صدق مي كنند. نتايج عددي به دست آمده نشان مي‌دهد كه اين روش‌ها براي حل اين نوع از مسائل موثر و كارا هستند.

the main disadavantages of the newton method are that the computation of the hessian matrix is a difficult 

امروزه مخاطرات زيست محيطي ناشي از سوخت­هاي فسيلي و همچنيندستيابي به منابع انرژي لازم و كافي براي توسعه به يكي از مهم­ترين دغدغه­­هايكشورها و دولت­مردان تبديل شده است؛ تا جايي­كه هر تحولي در حوزه انرژي تاثيراتبسزايي بر مناسبات بين­المللي مي­گذارد. در اين پژوهش، با تاكيد بر اثر شدتانرژي به­عنوان معيار كارايي انرژي به بررسي ارتباط بين رشد اقتصادي و مصرف انرژي دركشورهاي اوپك در فاصله زماني 2014-1980 پرداخته مي­شود. اين تحقيق با هدف بررسي اينكهآيا تأثير رشد اقتصادي بر مصرف انرژي با توجه به ميزان شدت انرژي متفاوت است يا خير،انجام شده است. اين مطالعه در ابتدا به بررسي رابطه علّي ميان رشد اقتصادي و مصرفانرژي مي­پردازد؛ نتايج وجود يك رابطه علّي يك­طرفه از رشد اقتصادي به مصرف انرژيرا در كشورهاي اوپك نشان مي­دهد. سپس با استفاده از يك مدل رگرسيون پانل آستانه­ايپويا، نقش و تأثير شدت انرژي بر رابطه بين مصرف انرژي و رشد اقتصادي كشورهاي عضواوپك بررسي مي­شود. مطابق با برآورد مدل رگرسيون پانل آستانه­اي پويا، سطح آستانه­ايشدت انرژي 27/7 برآورد شده است. در سطوح بالاتر از سطح آستانه­اي رشد اقتصادي تأثير مثبتو معني­داري برمصرف انرژي دارد. با اين حال ، در سطوح پايين­تر از سطح آستانه­اي رشد اقتصادي تأثيرمثبت و بي­معنايي بر مصرف انرژي داشته است. بدون در نظرگرفتن متغيرهاي كنترل، سطح آستانه­ايشدت انرژي 6/9 برآورد شده است.در سطوح بالاتر از سطح آستانه­اي رشد اقتصادي تأثير مثبت و معني­دارو در سطوح پايين­تر از سطح آستانه­اي، رشد اقتصادي تاثير منفي و بي­معني بر مصرفانرژي دارد.نتايج اين پژوهش براي سياست­گذاران انرژي و محيط زيست قابل توجه و اهميت است.

Diagonal quasi-Newton method 

Two spectral conjugate gradient methods based on some quasi-newton equation

The data of most real-world optimization problems (OPs) are often not known exactly at the same time the problem is being solved‎. ‎The reasons for data uncertainty contain measurement errors‎, ‎imprecise data‎, ‎future developments‎, ‎environmental conditions‎. ‎Thus‎, ‎using uncertain robust optimization for optimization problems with uncertain data is essential‎. ‎In robust optimization‎, ‎the uncertain parameters are assumed to belong to a set that is known prior‎, ‎and the focus lies on the worst case‎. ‎The goal is to ensure that the solution is feasible and works well in every possible future scenario‎. ‎An uncertain problem can be solved using the scalarization methods (Benson’s method and elastic constraint method) in multi objective optimization‎. ‎This thesis also focuses on a unified approach to characterizing different kinds of multi objective robustness concepts‎. ‎Based on linear and nonlinear scalarization results for several set order relations‎, ‎together with the help of image space analysis‎, ‎some suitable subsets of scalarization image space are introduced to make equivalent characterizations for upper set (lower set‎, ‎set‎, ‎certainly‎, ‎respectively) less ordered robustness for uncertain multi objective optimization problems‎. ‎In the sequel‎, ‎by virtue of scalar robust optimization and using a deterministic robust counterpart‎, ‎a more general form of the robust optimization is considered in which the objective function and constraints contains uncertain data‎. ‎Moreover‎, ‎the relation between uncertain optimization and the image set is analyzed‎. ‎This idea leads to solve a min-max problem‎. ‎Moreover‎, ‎several necessary and sufficient optimality conditions‎, ‎especially saddle point sufficient optimality conditions for scalar robust optimization problems‎, ‎are obtained‎. ‎Finally‎, ‎a simple example for finding a shortest path is included‎.

  In this thesis, an interactive method for solving discrete multi-criteria optimization problems is studied. This methods is based on the pairwise comparison of alternatives to obtain an optimal solution. It is assumed that there are p criteria, m alternatives and a single decision maker who has an implicit increasing quasi-concave utility (value) function that has to be optimized. The main point in this regard is designing a procedure   with lower number of comparison made by the decision make. To this aim, convex cones are used for ranking the alternatives. In the sequel, two methods based on the dual theory and evolutionary algorithms are introduced. These methods obtain an optimal solution by reducing the number of   pairwise comparison and using the decision maker's information.

  ‎‎‎In multiobjective optimization, several different objective should be optimized which are in conflict in general. Thus, the objective space is a set of vectors. For comparing these vectors a partial order is needed. Almost in all cases, this order is defined by a cone. This thesis studies multiobjective problems in which the related cone is a polyhedral cone. For polyhedral cones, it is shown how to find vectors in the positive dual cone that are needed for a scalarized objective function. Instructive examples are presented. In this thesis, by using a special kind of polyhedral cone namely, dilating cones and applying nonlinear scalarization proper are characterized. A similar characterization is derived for weakly efficient solution, for which no convexity hypotheses are required. Finally, when the feasible set is given by a cone constraint, some necessary and sufficient optimality conditions via a kind of scalar nonlinear Lagrangian are obtained.

  Quasi-newton methods are an important class of iterative methods for solving unconstrained optimization.These methods can be used when the Hessian matrix is difficult or time consuming to eraluate. Insted of using an estimate of the Hessian matrix, these methods buildup an approximate Hessian matrix by using gradient information. This family is one of the popular algorithms and has many advantages, but there are some drawbacks on them. This methods only use the gradient valuse and don't use the function valuse.Also, in most cases, the quasi newton method can't guarantee the positive definite of Hessian matrix.The modified quasi-newton methods suggest some proposals to overcoming this disadvantages.\\\\In this thesis, two new family of modified quasi-newton methods were investigated. firstly, a modified BFGS algorithm with global convergence properties for nonconvex functions were presented. In the second part by modifying the interpolation conditions to approximate the quadratic model of the function, a new modified quasi-newton equation is introduced. Based on this equation, a modified BFGS algorithm is presented.

  In this distribution after introducing the concept of expected shortfall (ES) as a financial risk measure we briefly discuss the coherence properties of this measure. This statistic risk measure arises in a natural way from the estimation of the average of the 100p percent worst losses in a sample of returns to a portfolio where p is some fixed confidence level. Then we comprehensively review several know parametric method for estimating expected shortfall.  

  Intervallinearprogrammingwasintroducedinordertodealwithlinearprogramming problems with uncertainties that are modelled by ranges admissible values. Basic tasks in interval linear programming such as calculating the optimal value bounds or set of all possible solutions may be comutationally very expensive. However, if some basis stability criterion holds true then the problems becomes much more easy to solve. We introduce a novel kind of robustness in linear programming. A solution x is called robust optimal if for all realizations of the objective function coe?cients and the constraint matrix entries from given interval domains there are aooropriate choices of the right-handside entriesfrom their intervaldomains suchthat x remains optimal. We propose a method to check for robustness of agiven point, and discuss topological properties of the robust optimal solution set. We illustrate applicability of our concept in tra  ortation and nutrition problems. Since note every problem has a robust optimal solution, we introduce also a concept of an approximate robust solution and develop an e?cient method. We discuss the problem of checking whether a given solution is optimal for each realization of interval data. This problem was studied for particular forms of linear programming problems. we extend the results to a general model and simplify the overall approach, Moreover, we i  ect coputational complexity, too. Eventually, we investigate a related optimality concept of semi-strong optimality.

روش هاي گراديان مزدوج يك خانواده مهم براي حل مسائل بهينه سازي نامقيد هستند. در اين روش ها به دليل عدم نياز به استفاده از ماتريس هسي يا تقريب آن، استفاده كم از حافظه ماشين و خواص همگرايي موضعي و سراسري مناسب، به روش هايي بسيار مطلوب براي حل مسائل بهينه سازي نامقيد در مقياس بزرگ تبديل شده اند. به دليل اين­كه اين الگوريتم ها به طور معمول فقط از اطلاعات مشتق مرتبه اول تابع هدف استفاده مي كنند، بنابران ممكن است همگرايي آنها كند باشد.در اين پايان نامه تركيبي از گراديان مزدوج مقياس يافته و روش مجانب متحرك براي حل مسائل بهينه سازي نامقيد غير خطي در مقياس بزرگ ارائه شده است.در اين روش­ها جهت كاهشي مورد استفاده در هر تكرار به وسيله حل زير مسايل جدايي­پذير محدب توليد شده توسط جهات­مجانبي توليد مي­گردد. همچنين در اين پايان نامه با استفاده از روش­هاي ناحيه اطمينان پارامتر­هاي مجانبي جديد و مؤثري تعريف مي­گردند. بررسي خواص همگرايي و عددي روش­ها هدف بعدي اين پايان نامه است.

  In this theisis, we describe an interactive procedural algorithm for convex multi- objective programming based upon the Tchebyche? method, Wierzbicki’s reference point approach, and the procedure of Michalowski and Szapiro. At each iteration, the decision maker (DM) has the option of expressing his or her objective-function aspirations in the form of a reference criterion vector. Also, the DM has the option of expressing minimally acceptable values for each of the objectives in the form of a reservationvector. Baseduponthisinformation, acertainregionisde?nedforexam- ination. In addition, a special set of weights is constructed. Then with the weights, the algorithm of this paper is able to generate a group of e?cient solutions that provides for an overall view of the current iteration’s certain region. By modi?cation of the reference and reservation vectors, one can ‘‘steer” the algorithm at each itera- tion. From a theoretical point of view, we prove that none of the e?cient solutions obtained using this scheme impair any reservation value for convex problems. The behavior of the algorithm is illustrated by means of graphical representations and an illustrative numerical example. we carry out an extension of the MICA method (modi?ed interactive chebyshev algorithm) for non-convex multiobjective programming. This method is based on the Tchebychev method and in the reference point approach. At each iteration, the decision maker (DM) can provide aspiration levels (desirable values for the objec- tive functions) and also, if the DM wishes, reservation levels (level under which the objective function is not considered acceptable). On the basis of this preferential in- formation, a region of the nondominated objective set is de?ned. In the convex case, considering the aspiration vector as a reference point in an achievement scalarizing function and taking a set of weight vectors, the e?cient solutions generated satisfy the reservation levels. In this work, we analyze the non-convex case. The main re- sult of MICA is veri?ed and demonstrated for the non-convex bi-objective case. The MICA method is not veri?ed in general for multiobjective problems with three or more objective functions, which is demonstrated with a counterexample. we concentrate on reference point based methods in multiobjective programming todemonstrate, asmaincontribution, thatthesolutiontoamultiobjectiveoptimiza- tion problem stays unchanged if the reference point is changed to any point on a set de?nedbymeansoftheoriginalreferencepoint,thenondominatedobjectivesolution and some parameters of the ASF. Concretely, this new set of “equivalent reference points” is the convex linear combination of two straight lines, one containing the original reference point and the other a nondominated objective solution, where the slope of both straight lines is given by the inverses of the weights of the ASF. An illustrative example is used to show the results obtained and an empirical model (application with real data) allows us to highlight possible implications.

  A trust-region-based algorithm for the nonconvex unconstrained multiobjective optimization problem is considered. It is a generalization of the algorithm proposed by Fliege et al. for convex problems. Similarly to the scalar case, at each iteration a subproblem is solved and the step needs to be evaluated. Therefore, the notions of decrease condition and of predicted reduction are adapted to the vectorial case. A rule to update the trust region radius is introduced. Under differentiability assump- tions, the algorithm converges to points satisfying a necessary condition for Pareto points and, in the convex case, to a Pareto points satisfying necessary and sufficient conditions. Furthermore, it is proved that the algorithm displays a q-quadratic rate of convergence. The global behavior of the algorithm is shown in the numerical ex- perience reported. Keyword: Multicriteria optimization, Multiobjective programming, Pareto points, Newton’s method ,Trust region.

  آنچه بهينه‌سازي را از كاربردهاي عملي دور نموده است ناشي از دو واقعيت عملي مهم است. يكي تنوع اهداف و معيارهاست كه در بيشتر موارد در تضاد و تقابل با يكديگرند و ديگري نادقيق بودن داده‌هاي اوليه است. براي رويارويي با اين چالش مفهوم پايداري در بهينه‌سازي چندهدفه ارائه شد كه مورد بحث اين پايان‌نامه است.  در اين پايان‌نامه، مفهوم پايداري از مسائل تك‌هدفه به مسائل چندهدفه تعميم داده مي‌شود. منظور از جواب پايدار جوابي است كه به‌ازاي تمام سناريوهاي ممكن كارا بماند. همچنين مفهوم پايداري مينماكس مورد مطالعه قرار مي‌گيرد. فرآيند ارائه شده در اين زمينه، تركيبي است از پايداري در مسائل تك‌هدفه و بهينه‌سازي چندهدفه معين. به‌طور خاص در مسائل بهينه‌سازي چندهدفه‌ي خطي، فرمولي براي شعاع پايداري شدني و شرايط بهينگي قابل اجرا براي جواب‌هاي كاراي ضعيف پايدار ارائه مي‌شود. ضمن معرفي جواب‌هاي كاراي ضعيف بسيار پايدار يك فرآيند محاسباتي نيز براي محاسبه‌ي اين جواب‌ها بررسي مي‌شود. سرانجام يك فرآيند بهينه‌سازي دوهدفه براي تقليل مرز كارا و كمك به تصميم‌گيرنده جهت اتخاذ تصميمي ارجح ارائه مي‌شود. در اين راستا، پايداري جواب و پايداري مدل مطرح مي‌شود. الگوريتم‌هاي محاسباتي مربوطه در اين زمينه براساس تجزيه‌ي وجهي است.

مسائل بهينه سازي ضربي دسته ي خاصي از مسائل بهينه سازي سراسري اند. الگوريتم هاي موجودبهينه سازي سراسري براي حل اين مسائل قابل اجرا هستند ولي كارايي پاييني دارند. اگر عوامل ضربدر بهينه سازي مثبت باشند، جواب بهينه ي اين مسائل يك جواب كارا براي يك مسأله ي بهينه سازيچندهدفه ي نظير است. بنابراين به جاي جستجوي سراسري روي كل فضاي شدني كافيست اين جوابرا در مرز كاراي مسأله ي چندهدفه ي مذكور كه يك مجموعه ي كوچكتر است جستجو كرد. در واقع الگوريتم هاي حل مسائل چندهدفه را مي تواند در اين راستا بسيار كارساز باشد. از جمله الگوريتم هايموثر در اين زمينه الگوريتم ارائه شده توسط ارگوت و شائو در سال 201? است [30]. در اين روشكه براي حل مسائل بهينه سازي چندهدفه ي محدب ارائه شده است، با يك روش برش و كران تقريبياز مرز كارا به دست مي آيد. ارگوت و همكاران[31] در سال 201? با تعميم روش تقريب بيرونيبنسون يك روش مشابه مبتني بر دوگان براي حل مسائل بهينه سازي خطي چندهدفه ارائه كردند. دراين پايانامه بر اساس اين روش الگوريتمي براي حل مسائل بهينه سازي خطي چندهدفه ي ضربي ارائهمي شود.

در اين پايان نامه مفاهيم بهينگي سره در بهينه سازي برداري با ساختارهاي ترتيبي متغير معرفي شده و با استفاده از برخي روشهاي اسكالرسازي جديد خواص مشخصه مختلفي براي تشخيص عناصر كاراي سره ارائه مي شود. اين اسكالرسازي ها براساس تابعكهايي تعريف ميشوند كه از عناصر مخروط دوگان افزوده، بدست مي آيند. ضمن بررسي رابطه ي بين مخروطهاي دوگان افزوده و مخروطهاي بيشاپ-فلپس، خواص اين تابعكها مورد مطالعه قرار ميگيرد. همچنين خواص مشخصه اي براي ديگر مفاهيم بهينگي مانند عناصر بهينه ضعيف و بهينه قوي بدست مي آيد..

‎The gradient methods family is an important family of existent method for solving unconstrained optimization problems‎. ‎The Barzilai-Borwein gradient‎‎method is one of the most important gradient methods that has low compution and appropriate speed convergence‎. ‎This method has R-superlinear convergence rate for two-dimensional strictly convex quadratic functions‎. ‎In this thesis‎, ‎we present a new convergence analysis for the BB gradient method that indicate this method has R-superlinear convergence rate with rate $\\sqrt{2}$‎. ‎In the second part of the thesis‎, ‎by combining the conjugate gradient methods and the BB method an algorithm useing of step size BB is presented that keeps appropriate properties of both methods‎. ‎the next topic of the thesis analyzes the convergence properties of the above-mentioned method‎.