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We consider a smooth pessimistic bilevel optimization problem, where the lower-level problem is convex and satisfies the Slater constraint qualification. These assumptions ensure that the Karush-Kuhn-Tucker (KKT) reformulation of our problem is well-defined. We then introduce and study the (i) Scholtes, (ii) Lin and Fukushima, (iii) Kadrani, Dussault and Benchakroun, (iv) Steffensen and Ulbrich, and (v) Kanzow and Schwartz relaxation methods for the KKT reformulation of our pessimistic bilevel program. These relaxations have been extensively studied and compared for mathematical programs with complementatrity constraints (MPCCs). To the best of our knowledge, such a study has not been conducted for the pessimistic bilevel optimization problem, which is completely different from an MPCC, as the complemetatrity conditions are part of the objective function, and not in the feasible set of the problem. After introducing these relaxations, we provide convergence results for global and local optimal solutions, as well as suitable versions of the C- and M-stationarity points of our pessimistic bilevel optimization problem. Numerical results are also provided to illustrate the practical implementation of these relaxation algorithms, as well as some preliminary comparisons., Comment: 24 pages, 4 figures, 2 tables

The aim of this paper is to study the weak convergence analysis of sequence of iterates generated by a three-operator splitting method of Davis and Yin incorporated with two-step inertial extrapolation for solving monotone inclusion problem involving the sum of two maximal monotone operators and a co-coercive operator in Hilbert spaces. Our results improve on the setbacks observed recently in the literature that one-step inertial Douglas-Rachford splitting method may fail to provide acceleration. Our convergence results also dispense with the summability conditions imposed on inertial parameters and the sequence of iterates assumed in recent results on multi-step inertial methods in the literature. Numerical illustrations from image restoration problem and Smoothly Clipped Absolute Deviation (SCAD) penalty problem are given to show the efficiency and advantage gained by incorporating two-step inertial extrapolation over one-step inertial extrapolation for three-operator splitting method.

Generative diffusion models have recently emerged as a powerful strategy to perform stochastic sampling in Bayesian inverse problems, delivering remarkably accurate solutions for a wide range of challenging applications. However, diffusion models often require a large number of neural function evaluations per sample in order to deliver accurate posterior samples. As a result, using diffusion models as stochastic samplers for Monte Carlo integration in Bayesian computation can be highly computationally expensive. This cost is especially high in large-scale inverse problems such as computational imaging, which rely on large neural networks that are expensive to evaluate. With Bayesian imaging problems in mind, this paper presents a Multilevel Monte Carlo strategy that significantly reduces the cost of Bayesian computation with diffusion models. This is achieved by exploiting cost-accuracy trade-offs inherent to diffusion models to carefully couple models of different levels of accuracy in a manner that significantly reduces the overall cost of the calculation, without reducing the final accuracy. The effectiveness of the proposed Multilevel Monte Carlo approach is demonstrated with three canonical computational imaging problems, where we observe a $4\times$-to-$8\times$ reduction in computational cost compared to conventional Monte Carlo averaging., Comment: 13 images

This work develops a particle system addressing the approximation of McKean-Vlasov stochastic differential equations (SDEs). The novelty of the approach lies in involving low discrepancy sequences nontrivially in the construction of a particle system with coupled noise and initial conditions. Weak convergence for SDEs with additive noise is proven. A numerical study demonstrates that the novel approach presented here doubles the respective convergence rates for weak and strong approximation of the mean-field limit, compared with the standard particle system. These rates are proven in the simplified setting of a mean-field ordinary differential equation in terms of appropriate bounds involving the star discrepancy for low discrepancy sequences with a group structure, such as Rank-1 lattice points. This construction nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An asymptotic error analysis reveals that the proposed approach outperforms methods based on the classic particle system with independent initial conditions and noise.

Durum wheat has been cultivated since the beginning of crop domestication, occupying now the tenth ranking among the global most significant cultivated crops. Despite the fact that, the extent of the crop genetic diversity has not yet fully incorporated into modern varieties through breeding programs. In this study, a total of 35 markers (11 RAPD, 12 ISSR, and 12 CDDP) were utilized to assess the genetic variability and population structure of sixteen different cultivars of Iraqi durum wheat. Out of 294 bands obtained, 171 were identified as polymorphic: 47.00 polymorphic alleles from 98 RAPD bands, 53 polymorphic alleles from a total of 89 ISSR bands, and 71 alleles from 107 CDDP bands. The average number of observed alleles (Na), effective number of alleles (Ne), Shannon's information index (I), expected heterozygosity or gene diversity (He), unbiased expected heterozygosity (uHe), and polymorphic information content (PIC) (1.45, 1.38, 0.32, 0.22, 0.24, and 0.28, respectively) were obtained for RAPDs , (1.63, 1.45, 0.40, 0.27, 0.29, and 0.32, respectively) ISSRs and (1.35, 1.35, 0.31, 0.21, 0.23, and 0.30, respectively) for the CDDP markers. A dendrogram of two main clades (unweighted pair group method with arithmetic mean; UPGMA) and three populations of structure analysis, were obtained based on the three markers data. The analysis of molecular variance indicated 97.00%, 97.00%, and 90.00% variability within populations, applying RAPD, ISSR, and CDDP markers, respectively. The highest diversity indices were revealed in population 2 under the RAPD and CDDP markers, whereas population 1 had the highest values of these indices according to the ISSR markers. The results provide greater knowledge on the genetic makeup of Iraqi durum wheat cultivars, that facilitate future breeding programs of this crop., Comment: 15

In this work, we propose some new Douglas-Rashford splitting algorithms for solving a class of generalized DC (difference of convex functions) in real Hilbert spaces. The proposed methods leverage the proximal properties of the nonsmooth component and a fasten control parameter which improves the convergence rate of the algorithms. We prove the convergence of these methods to the critical points of nonconvex optimization under reasonable conditions. We evaluate the performance and effectiveness of our methods through experimentation with three practical examples in machine learning. Our findings demonstrated that our methods offer efficiency in problem-solving and outperform state-of-the-art techniques like the DCA (DC Algorithm) and ADMM.

Many recently proposed gradient projection algorithms with inertial extrapolation step for solving quasi-variational inequalities in Hilbert spaces are proven to be strongly convergent with no linear rate given when the cost operator is strongly monotone and Lipschitz continuous. In this paper, our aim is to design an inertial type gradient projection algorithm for quasi-variational inequalities and obtain its linear rate of convergence. Therefore, our results fill in the gap for linear convergence results for inertial type gradient projection algorithms for quasi variational inequalities in Hilbert spaces. We perform numerical implementations of our proposed algorithm and give numerical comparisons with other related inertial type gradient projection algorithms for quasi variational inequalities in the literature.

The Extreme Learning Machine (ELM) technique is a machine learning approach for constructing feed-forward neural networks with a single hidden layer and their models. The ELM model can be constructed while being trained by concurrently reducing both the modeling errors and the norm of the output weights. Usually, the squared loss is widely utilized in the objective function of ELM, which can be treated as a LASSO problem and hence applying the fast iterative shrinkage thresholding algorithm (FISTA). However, in this paper, the minimization problem is solved from a variational inequalities perspective giving rise to improved algorithms that are more efficient than the FISTA. A fast general extra-gradient algorithm which is a form of first-order algorithm is developed with inertial acceleration techniques and variable stepsize which is updated at every iteration. The strong convergence and linear rate of convergence of the proposed algorithm are established under mild conditions. In terms of experiments, two experiments were presented to illustrate the computational advantage of the proposed method to other algorithms., Comment: 24 pages