We prove convergence of Anderson acceleration for a class of nonsmooth fixed-point problems for which the nonlinearities can be split into a smooth contractive part and a nonsmooth part which has a small Lipschitz con...
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We prove convergence of Anderson acceleration for a class of nonsmooth fixed-point problems for which the nonlinearities can be split into a smooth contractive part and a nonsmooth part which has a small Lipschitz constant. These problems arise from compositions of completely continuous integral operators and pointwise nonsmooth functions. We illustrate the results with two examples.
Real life problems are used as benchmarks to evaluate the performance of existing, improved and modified evolutionary algorithms. In this paper, we propose a new hybrid method, namely SIWO, by embedding space transfor...
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Real life problems are used as benchmarks to evaluate the performance of existing, improved and modified evolutionary algorithms. In this paper, we propose a new hybrid method, namely SIWO, by embedding space transformation search (STS) into invasive weed optimization to solve complex fixed-point problems. Invasive weed optimization suffers from premature convergence when solving complex optimization problems. Using STS transforms the current search space into a new search space by simultaneously evaluating solutions in the current and transformed spaces. This increases the probability that a solution is closer to the global optimum. Therefore, we can avoid premature convergence and the convergence speed is also increased. To evaluate the performance of SIWO, four complex fixed-point problems are chosen from the literature. Our findings demonstrate that SIWO can solve complex fixed-point problems with great precision. Moreover, the numerical results demonstrate that SIWO is an effective and efficient algorithm compared with some state-of-the-art algorithms.
This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixedpointproblems defined in a Hilbert space. A general strong convergence theo...
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This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixedpointproblems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixedpoint iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.
We introduce a machine-learning framework to warm-start fixed-point optimization algorithms. Our architecture consists of a neural network mapping problem parameters to warm starts, followed by a predefined number of ...
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We introduce a machine-learning framework to warm-start fixed-point optimization algorithms. Our architecture consists of a neural network mapping problem parameters to warm starts, followed by a predefined number of fixed-point iterations. We propose two loss functions designed to either minimize the fixed-point residual or the distance to a ground truth solution. In this way, the neural network predicts warm starts with the end-to-end goal of minimizing the downstream loss. An important feature of our architecture is its exibility, in that it can predict a warm start for fixed-point algorithms run for any number of steps, without being limited to the number of steps it has been trained on. We provide PAC-Bayes generalization bounds on unseen data for common classes of fixed-point operators: contractive, linearly convergent, and averaged. Applying this framework to well-known applications in control, statistics, and signal processing, we observe a significant reduction in the number of iterations and solution time required to solve these problems, through learned warm starts.
In this paper, we propose a new modified proximal point algorithm for a finite family of non-expansive mappings in the framework of CAT(0) spaces. We establish -convergence and strong convergence theorems under some m...
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In this paper, we propose a new modified proximal point algorithm for a finite family of non-expansive mappings in the framework of CAT(0) spaces. We establish -convergence and strong convergence theorems under some mild conditions. Our results extend some known results which appeared in the literature.
Recently, there have been numerous insightful applications of zero-sum stochastic differential games in insurance, as discussed in Liu et al. [Liu, J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an i...
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Recently, there have been numerous insightful applications of zero-sum stochastic differential games in insurance, as discussed in Liu et al. [Liu, J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer with regime-switching and risk constraint. Scandinavian Actuarial Journal 2014(7), 583-601]. While there could be some practical situations under which nonzero-sum game approach is more appropriate, the development of such approach within actuarial contexts remains rare in the existing literature. In this article, we study a class of nonzero-sum reinsurance-investment stochastic differential games between two competitive insurers subject to systematic risks described by a general compound Poisson risk model. Each insurer can purchase the excess-of-loss reinsurance to mitigate both systematic and idiosyncratic jump risks of the inter-arrival claims;and can invest in one risk-free asset and one risky asset whose price dynamics follows the famous Heston stochastic volatility model [Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies6, 327-343]. The main objective of each insurer is to maximize the expected utility of his terminal surplus relative to that of his competitor. Dynamic programming principle for this class of nonzero-sum game problems leads to a non-canonical fixed-point problem of coupled non-linear integral-typed equations. Despite the complex structure, we establish the unique existence of the Nash equilibrium reinsurance-investment strategies and the corresponding value functions of the insurers in a representative example of the constant absolute risk aversion insurers under a mild, time-independent condition. Furthermore, Nash equilibrium strategies and value functions admit closed forms. Numerical studies are also provided to illustrate the impact of the systematic risks on the Nash equilibrium strategies. Finally, we connect our
In this paper, we consider parametric primal and dual equilibrium problems in locally convex Hausdorff topological vector spaces. Sufficient conditions for the approximate solution maps to be Hausdorff continuous are ...
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In this paper, we consider parametric primal and dual equilibrium problems in locally convex Hausdorff topological vector spaces. Sufficient conditions for the approximate solution maps to be Hausdorff continuous are established. We provide many examples to illustrate the essentialness of the imposed assumptions. As applications of these results, the Hausdorff continuity of the approximate solution maps for optimization problems, variational inequalities and fixed-point problems are derived.
The Bregman function-based Proximal point Algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additiona...
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The Bregman function-based Proximal point Algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additional feature consists in an application of zone coercive regularizations. The latter allows to treat the generated subproblems as unconstrained ones, albeit with a certain precaution in numerical experiments. However, compared to the (classical) Proximal point Algorithm for equilibrium problems, convergence results require additional assumptions which may be seen as the price to pay for unconstrained subproblems. Unfortunately, they are quite demanding - for instance, as they imply a sort of unique solvability of the given problem. The main purpose of this paper is to develop a modification of the BPPA, involving an additional extragradient step with adaptive (and explicitly given) stepsize. We prove that this extragradient step allows to leave out any of the additional assumptions mentioned above. Hence, though still of interior proximal type, the suggested method is applicable to an essentially larger class of equilibrium problems, especially including non-uniquely solvable ones.
Conjugate heat-transfer problems are typically solved using partitioned methods where fluid and solid subdomains are evaluated separately by dedicated solvers coupled through a common boundary. Strongly coupled scheme...
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Conjugate heat-transfer problems are typically solved using partitioned methods where fluid and solid subdomains are evaluated separately by dedicated solvers coupled through a common boundary. Strongly coupled schemes for transient analysis require fluid and solid problems to be solved many times each time step until convergence to a steady state. In many practical situations, a fairly simple and frequently employed fixed-point iteration process is rather ineffective;it leads to a large number of iterations per time step and consequently to long simulation times. In this article, Anderson mixing is proposed as a fixed-point convergence acceleration technique to reduce computational cost of thermal coupled fluidsolid problems. A number of other recently published methods with applications to similar fluidstructure interaction problems are also reviewed and analyzed. Numerical experiments are presented to illustrate relative performance of these methods on a test problem of rotating pre-swirl cavity air flow interacting with a turbine disk. It is observed that performance of Anderson mixing method is superior to that of other algorithms in terms of total iteration counts. Additional computational savings are demonstrated by reusing information from previously solved time steps. Copyright (c) All rights reserved 2012 Rolls-Royce plc.
A general linear boundary value problem for a nonlinear system of delay differential equations (DDE in short) is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unk...
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A general linear boundary value problem for a nonlinear system of delay differential equations (DDE in short) is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by piecewise linear function v(h) defined by its values v(i) = v(h)(t(i)) at grid points t(i), i = 0, 1, ... , N, where N is a given positive integer and h = max(1 <= i <= N)(t(i) - t(i-1)). Under suitable assumptions, the existence of a fixed-point of A is equivalent to existence of so called epsilon(h)-approximate fixed-points of v(h) = Av(h), which can be found by minimization of L-2((n))) norm of residuum v(h) - Av(h), with respect to the variables v(i) These epsilon(h)-approximate fixed-points are used for obtaining approximate solutions of the original boundary value problem for a system of DDE. Numerical experiments with the boundary value problem for a system of delay differential equations of population dynamics as well as with two periodic boundary value problems: one for the periodic distributed delay Lotka-Volterra competition system and the second one modeling two coupled identical neurons with time-delayed connections show an efficiency of this kind of approach. (C) 2011 Elsevier B.V. All rights reserved.
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