proximal method
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2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Fouzia Amir ◽  
Ali Farajzadeh ◽  
Jehad Alzabut

Abstract Multiobjective optimization is the optimization with several conflicting objective functions. However, it is generally tough to find an optimal solution that satisfies all objectives from a mathematical frame of reference. The main objective of this article is to present an improved proximal method involving quasi-distance for constrained multiobjective optimization problems under the locally Lipschitz condition of the cost function. An instigation to study the proximal method with quasi distances is due to its widespread applications of the quasi distances in computer theory. To study the convergence result, Fritz John’s necessary optimality condition for weak Pareto solution is used. The suitable conditions to guarantee that the cluster points of the generated sequences are Pareto–Clarke critical points are provided.


Author(s):  
Spyridon Pougkakiotis ◽  
Jacek Gondzio

AbstractIn this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in Pougkakiotis and Gondzio (Comput Optim Appl 78:307–351, 2021. 10.1007/s10589-020-00240-9) for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM.


2021 ◽  
Vol 36 ◽  
pp. 04007
Author(s):  
Gillian Yi Han Woo ◽  
Hong Seng Sim ◽  
Yong Kheng Goh ◽  
Wah June Leong

In this paper, we propose to use spectral proximal method to solve sparse optimization problems. Sparse optimization refers to an optimization problem involving the ι0 -norm in objective or constraints. The previous research showed that the spectral gradient method is outperformed the other standard unconstrained optimization methods. This is due to spectral gradient method replaced the full rank matrix by a diagonal matrix and the memory decreased from Ο(n2) to Ο(n). Since ι0-norm term is nonconvex and non-smooth, it cannot be solved by standard optimization algorithm. We will solve the ι0 -norm problem with an underdetermined system as its constraint will be considered. Using Lagrange method, this problem is transformed into an unconstrained optimization problem. A new method called spectral proximal method is proposed, which is a combination of proximal method and spectral gradient method. The spectral proximal method is then applied to the ι0-norm unconstrained optimization problem. The programming code will be written in Python to compare the efficiency of the proposed method with some existing methods. The benchmarks of the comparison are based on number of iterations, number of functions call and the computational time. Theoretically, the proposed method requires less storage and less computational time.


Author(s):  
Spyridon Pougkakiotis ◽  
Jacek Gondzio

Abstract In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.


Author(s):  
Yana I. Vedel ◽  
Vladimir V. Semenov ◽  
Kateryna M. Golubeva

We propose a novel two-step proximal method for solving equilibrium problems in Hadamard spaces. The equilibrium problem is very general in the sense that it includes as special cases many applied mathematical models such as: variational inequalities, optimization problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithm is the analog of the two-step algorithm for solving the equilibrium problem in Hilbert spaces explored earlier. We prove the weak convergence of the sequence generated by the algorithm for pseudo-monotone bifunctions. Our results extend some known results in the literature for pseudo-monotone equilibrium problems.


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