Nondifferentiable Optimization Problems Research Articles

Parallel Algorithm of Balanced Sparse Packing of Rectangular Parallelepipeds

Introduction. Varieties of the problem of packing of rectangular parallelepipeds have a wide practical application in various fields of activity, for example, in the optimal filling of containers, in the design and layout of a wide variety of technological objects and systems, in the creation of backup copies on removable media, in the optimization of storage, protection and transportation of goods, in additive manufacturing, etc. This work continues research on this topic and is devoted to the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds of different sizes into a rectangular parallelepiped of minimum volume. It presents a mathematical model for this packing problem and a parallel algorithm for solving it. This algorithm is based on the reduction of the original problem to an unconditional optimization problem using penalty functions, which is solved by the multistart method, in which r-algorithm is used to find local minima from the set of generated starting points. The purpose. Construction of a mathematical model and development of an algorithm for solving the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume. Results. A mathematical model and a parallel algorithm for balanced sparse packing of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume are presented. The algorithm is based on reducing the problem with the help of penalty functions to an unconditional nondifferentiable optimization problem, for finding the solution of which multistart method is used in combination with r-algorithm for finding local minima. The results of numerical experiments are given. Conclusions. The application of the algorithm described in the work based on non-smooth optimization methods allows to improve the results of balanced sparse packing of rectangular parallelepipeds in an acceptable time. Numerical experiments showed effectiveness of the algorithm in practice. Keywords: balanced sparse packing, multistart method, r-algorithm, penalty function, "Master-Slave" procedure, numerical experiments.

Vision Transformer-Based Video Hashing Retrieval for Tracing the Source of Fake Videos

With the increasing negative impact of fake videos on individuals and society, it is crucial to detect different types of forgeries. Existing forgery detection methods often output a probability value, which lacks interpretability and reliability. In this paper, we propose a source-tracing-based solution to find the original real video of a fake video, which can provide more reliable results in practical situations. However, directly applying retrieval methods to traceability tasks is infeasible since traceability tasks require finding the unique source video from a large number of real videos, while retrieval methods are typically used to find similar videos. In addition, training an effective hashing center to distinguish similar real videos is challenging. To address the above issues, we introduce a novel loss function, hash triplet loss, to capture fine-grained features with subtle differences. Extensive experiments show that our method outperforms state-of-the-art methods on multiple datasets of object removal (video inpainting), object addition (video splicing), and object swapping (face swapping), demonstrating excellent robustness and cross-dataset performance. The effectiveness of the hash triplet loss for nondifferentiable optimization problems is validated through experiments in similar video scenes.

Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path” where the discontinuous ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-function provides the exact sparsity count; the “ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function provides a convex surrogate of sparsity count; and the “capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the nonconvex nondifferentiable capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function aims to enhance the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Our study of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-regularized problem offers interesting theoretical properties and practical compromise between the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path and the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Indeed, while the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function.

A particle swarm inspired approach for continuous distributed constraint optimization problems

Distributed Constraint Optimization Problems (DCOPs) are a widely studied framework for coordinating interactions in cooperative multi-agent systems. In classical DCOPs, variables owned by agents are assumed to be discrete. However, in many applications, such as target tracking or sleep scheduling in sensor networks, continuous-valued variables are more suitable than discrete ones. To better model such applications, researchers have proposed Continuous DCOPs (C-DCOPs), an extension of DCOPs, that can explicitly model problems with continuous variables. The state-of-the-art approaches for solving C-DCOPs experience either onerous memory or computation overhead and are unsuitable for non-differentiable optimization problems. To address this issue, we propose a new C-DCOP algorithm, namely Particle Swarm Optimization Based C-DCOP (PCD), which is inspired by Particle Swarm Optimization (PSO), a well-known centralized population-based approach for solving continuous optimization problems. In recent years, population-based algorithms have gained significant attention in classical DCOPs due to their ability in producing high-quality solutions. Nonetheless, to the best of our knowledge, this class of algorithms has not been utilized to solve C-DCOPs and there has been no work evaluating the potential of PSO in solving classical DCOPs or C-DCOPs. In light of this observation, we adapted PSO, a centralized algorithm, to solve C-DCOPs in a decentralized manner. The resulting PCD algorithm not only produces good-quality solutions but also finds solution without any requirement for derivative calculations. Moreover, we design a crossover operator that can be used by PCD to further improve the quality of solutions found. Finally, we theoretically prove that PCD is an anytime algorithm and empirically evaluate PCD against the state-of-the-art C-DCOP algorithms in a wide variety of benchmarks.

A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks

To fit sparse linear associations, a LASSO sparsity inducing penalty with a single hyperparameter provably allows to recover the important features (needles) with high probability in certain regimes even if the sample size is smaller than the dimension of the input vector (haystack). More recently learners known as artificial neural networks (ANN) have shown great successes in many machine learning tasks, in particular fitting nonlinear associations. Small learning rate, stochastic gradient descent algorithm and large training set help to cope with the explosion in the number of parameters present in deep neural networks. Yet few ANN learners have been developed and studied to find needles in nonlinear haystacks. Driven by a single hyperparameter, our ANN learner, like for sparse linear associations, exhibits a phase transition in the probability of retrieving the needles, which we do not observe with other ANN learners. To select our penalty parameter, we generalize the universal threshold of Donoho and Johnstone (Biometrika 81(3):425–455, 1994) which is a better rule than the conservative (too many false detections) and expensive cross-validation. In the spirit of simulated annealing, we propose a warm-start sparsity inducing algorithm to solve the high-dimensional, non-convex and non-differentiable optimization problem. We perform simulated and real data Monte Carlo experiments to quantify the effectiveness of our approach.

On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities

In this article, we study nonsmooth interval-valued multiobjective programming problem and generalized Stampacchia vector variational inequality with its weak form for interval-valued functions. Using the tools of Mordukhovich subdifferential, we define some new classes of generalized approximate LU-convex functions. These functions are then employed to establish the relations between the solutions of generalized vector variational inequalities and the approximate LU-efficient solutions of the nonsmooth interval-valued multiobjective programming problem. Moreover, we identify the Kuhn–Tucker vector critical points of the considered nonsmooth interval-valued multiobjective programming problem. Under suitable constraint qualification, we establish the equivalence among local approximate LU-efficient points, Kuhn–Tucker vector critical points and the solutions of generalized Stampacchia vector variational inequalities. The results of this paper extend and sharpen the corresponding results of [Giannessi F. On Minty variational principle. In: Giannessi F, Komlósi S, Rapcśak T. editors. New trends in mathematical programming. Dordrecht: Kluwer Academic Publishers; 1997. p. 93–99], [Golestani M, Sadeghi H, Tavan Y. Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi-efficiency. J Optim Theory Appl. 2018;179(3):896–916], [Lee GM, Lee KB. Vector variational inequalities for nondifferentiable convex vector optimization problems. J Global Optim. 2005;32(4):597–612], [Mishra SK, Upadhyay BB. Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity. 2013;17(4):1071–1083], [Upadhyay BB, Mohapatra RN, Mishra SK. On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv Nonlinear Var Inequal. 2017;20(2):1–12], [Zhang J, Zheng Q,Ma X, Li L. Relationships between interval-valued vector optimization problems and vector variational inequalities. Fuzzy Optim Decis Mak. 2016;15(1):33–55] for nonsmooth interval-valued multiobjective programming problem by using the powerful tool of Mordukhovich subdifferential.

Emergency Service Location Problem with Ring Roads

A key purpose of a service network design is to determine the best possible location(s) for each service center. The facility services must provide a quick and easy response to callers within a reasonable distance, especially in urgent cases. It’s well known that this problem is NP-hard, non-convex and non-differentiable optimization problem. However, if we make some simplifying assumptions, the problem could be solved within a polynomial time. This article introduces an algorithm for solving the problem of determining the best possible locations of the emergency service centers, concerning the case in which these centers are located on simple closed curves (e.g., ring roads). The proposed model can be applied in designing the emergency centers on ring roads in new cities, which have become one of the most important designs in solving traffic congestion problems. The problem is mathematically formulated, an algorithm for solving the problem under simplifying assumptions is proposed, the mathematics behind the algorithm is given, and the algorithm is illustrated by a numerical example.

Imbalanced Sample Selection With Deep Reinforcement Learning for Fault Diagnosis

An imbalanced number of faulty and normal samples causes serious damage to the performance of the conventional diagnosis methods. To settle the data-imbalance fault diagnosis problem, this article presents a novel general imbalanced sample selection strategy (DiagSelect) based on deep reinforcement learning. In DiagSelect, the problem of imbalanced sample selection from the training set is formulated as a multiarmed bandit problem of deep reinforcement learning. The nondifferentiable optimization problem of imbalanced sample selection can be solved by the Markov decision process. The parameters of DiagSelect can be optimized by REINFORCE with the feedback of the validation set. DiagSelect performs intelligent imbalanced sample selection to obtain better diagnosis performance autonomously. As a data-level technique, DiagSelect can be easily used in conjunction with the conventional diagnosis models. DiagSelect is validated in a synthetic dataset and an industrial process dataset. The results have shown the effectiveness, stability, and transferability of DiagSelect.

Linex-RSVM: Ramp Linex Support Vector Machine

We propose a novel support vector machine named Ramp Linex Support Vector Machine (Linex-RSVM) for classification problems. By introducing the ramp Linex loss function, the Linex-RSVM has several advantages compared with existing SVMs. First of all, the Ramp Linex loss function is bounded so that the Linex-RSVM can avoid the influence of outliers while trading. Secondly, Linex-RSVM has the sparseness, which is the most essential characteristic of C-SVM, because of the introduction of ∈-insensitive region. Lastly, the Linex-RSVM treats instances differently based on the location of each point, it gives heavier penalty to the instances between the two support hyperplanes and gives lighter penalty to other instances. We adapt the Concave-Convex Procedure (CCCP) to solve the non-convex and non-differential optimization problem in Linex-RSVM and demonstrate effectiveness on numerical experiments.

An improved algorithm for high-dimensional continuous threshold expectile model with variance heterogeneity

In the era of big data, structural variation and heterogeneity of high-dimensional data bring a great challenge to statistical modelling and computation. In this paper, wepropose a high-dimensional continuous threshold model for such data and develop a regularized expectile framework with the non-convex SCAD penalty. Due to theexistence of threshold, we have to tackle a non-linear and non-differentiable optimization problem. Inspired by the linear reparameterization technique and LLA strategy,we propose an efficient algorithm to solve this problem. Theoretical analysis and simulation results show that compared with the classical grid-search method, ouralgorithm has considerable computational advantages. Besides, our proposed method also performs well in variable selection, parameter estimation, thresholdidentification and data heterogeneity detection. These results indicate that our proposed method is effective in analysing high-dimensional heterogeneous data.

Photonic Band Gap Material Topological Design at Specified Target Frequency

AbstractTo achieve any specified band gap structure around target frequency, a new photonic crystals (PhC) topological design strategy is proposed. In the design optimization model, a band gap index that measures the minimal signed distances between the specified frequency and any two adjacent orders of frequencies is proposed, and the topology of PhC unit cell is represented with a low number of design variables through the material‐field series‐expansion. Then the sequential Kriging‐based algorithm is adopted for solving the multi‐peak and nondifferentiable optimization problems. Based on the numerical implementation of the proposed optimization method, the PhC unit cells are designed to provide band gaps precisely matching any specified central frequency value. The optimized unit cell designs are easily assembled to achieve a PhC device with the large relative band gap.

Inverse reinforcement learning in contextual MDPs

We consider the task of Inverse Reinforcement Learning in Contextual Markov Decision Processes (MDPs). In this setting, contexts, which define the reward and transition kernel, are sampled from a distribution. In addition, although the reward is a function of the context, it is not provided to the agent. Instead, the agent observes demonstrations from an optimal policy. The goal is to learn the reward mapping, such that the agent will act optimally even when encountering previously unseen contexts, also known as zero-shot transfer. We formulate this problem as a non-differential convex optimization problem and propose a novel algorithm to compute its subgradients. Based on this scheme, we analyze several methods both theoretically, where we compare the sample complexity and scalability, and empirically. Most importantly, we show both theoretically and empirically that our algorithms perform zero-shot transfer (generalize to new and unseen contexts). Specifically, we present empirical experiments in a dynamic treatment regime, where the goal is to learn a reward function which explains the behavior of expert physicians based on recorded data of them treating patients diagnosed with sepsis.

Discrete variable optimization of structures subjected to dynamic loads using equivalent static loads and metaheuristic algorithms

This paper presents a new computational procedure for optimization of structures subjected to dynamic loads. The optimization problem is formulated with discrete design variables that represent the members from a table of commercially available members. Also, the requirements in the American Institute of Steel Construction (AISC) manual are formulated as constraints. This results in a nondifferentiable optimization problem. In the new procedure, the dynamic load is transformed into equivalent static loads (ESLs). Then the static response optimization problem having discrete design variables is solved using a metaheuristic optimization algorithm. Three methods to calculate the ESLs are investigated. It is found that the ESL cycles cannot converge to the final design. Therefore after a few ESL cycles, the original dynamic loads need to be used in the optimization process. Four example problems are solved to analyze the procedure. Based on this analysis, it is concluded that the new procedure is more efficient compared to a procedure that does not use the ESL cycles because it reduces the total CPU effort to obtain the final design. Also, better final designs are found. The reason is that many more designs are analyzed very efficiently with the ESL procedure.

Inexact Cuts in Stochastic Dual Dynamic Programming Applied to Multistage Stochastic Nondifferentiable Problems

In [V. Guigues, SIAM J. Optim., 30 (2020), pp. 407--438], an inexact variant of stochastic dual dynamic programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the method. That variant of SDDP was studied in [V. Guigues, SIAM J. Optim., 30 (2020), pp. 407--438] for linear and for differentiable nonlinear multistage stochastic programs (MSPs). In this paper, we extend ISDDP to nondifferentiable MSPs. We first provide formulas for inexact cuts for value functions of convex nondifferentiable optimization problems. We then combine these cuts with SDDP to describe ISDDP for nondifferentiable MSPs and analyze the convergence of the method. More precisely, for a problem with $T$ stages, we show that for errors bounded from above by $\varepsilon$, the limit superior and limit inferior of sequences of upper and lower bounds on the optimal value of the problem are at most at distance $3 \varepsilon T$ to the optimal value and that for asymptotically vanishing errors ISDDP converges to an optimal policy. Finally, we present the results of encouraging numerical experiments on a multistage nondifferentiable stochastic convex program solved using exact SDDP and the proposed ISDDP.

A proximal gradient splitting method for solving convex vector optimization problems

In this paper, a proximal gradient splitting method for solving nondifferentiable vector optimization problems is proposed. The convergence analysis is carried out when the objective function is the sum of two convex functions where one of them is assumed to be continuously differentiable. The proposed splitting method exhibits full convergence to a weakly efficient solution without assuming the Lipschitz continuity of the Jacobian of the differentiable component. To carry this analysis, the popular Beck–Teboulle's line-search procedure is extended to the vectorial setting under mild assumptions. It is also shown that the proposed scheme obtains an ϵ-approximate solution to the vector optimization problem in at most iterations.

Ramp loss for twin multi-class support vector classification

ABSTRACTTwin K-class support vector classification (TKSVC) adopts ‘One-vs.-One-vs.-Rest’ structure to utilise all the samples to increase the prediction accuracy. However, TKSVC is sensitive to noises or outliers due to the use of the Hinge loss function. To reduce the negative influence of outliers, in this paper, we propose a more robust algorithm termed as Ramp loss for twin K-class support vector classification (Ramp-TKSVC) where we use the Ramp loss function to substitute the Hinge loss function in TKSVC. Because the Ramp-TKSVC is a non-differentiable non-convex optimisation problem, we adopt Concave–Convex Procedure (CCCP) to solve it. To overcome the drawbacks of conventional multi-classification methodologies, the TKSVC is utilised as a core of our Ramp-TKSVC. In the Ramp-TKSVC, the outliers are prevented from becoming support vectors, thus they are not involved in the construction of hyperplanes, making the Ramp-TKSVC more robust. Besides, the Ramp-TKSVC is sparser than the TKSVC. To verify the validity of our Ramp-TKSVC, we conduct experiments on 12 benchmark datasets in both linear and nonlinear cases. The experimental results indicate that our algorithm outperforms the other five compared algorithms.

An exact penalty method for constrained optimal control problems

We develop a theoretical framework for constrained optimal control problems. Using an exact penalty method, the differential equation as well as the constraints are handled to obtain an abstract nonsmooth infinite dimensional optimization problem. By tools of nonsmooth analysis theory in infinite dimensional spaces, we investigate the relation between the stationary points of the obtained nondifferentiable optimization problem and those of the constrained optimal control problem. An algorithm to minimize the unconstrained problem combined the penalty function and the projection method is also given.

Optimality conditions for an exhausterable function on an exhausterable set

Exhausters are families of convex compact sets that allow one to represent directional derivative of the studied function at a point in the form of InfMax or SupMin of linear functions. Functions for which such a representation is valid we call exhausterable. This class of functions is quite wide and contains many nonsmooth ones. The set of exhausterable function is also called exhausterable. In the present paper we describe optimality conditions for an exhausterable function on an exhausterable set. These conditions can be used for solving many nondifferentiable optimization problems. An example that illustrate obtained results is provided.

Модифицированный метод симплексных погружений для решения задач выпуклой оптимизации с большим числом ограничений

Рассматривается метод симплексных погружений, адаптированный для решения задач выпуклой оптимизации с большим числом ограничений. Разработаны две модификации, позволяющие ускорять работу метода. Первая из них использует более экономичный способ расчета невязок ограничений, что позволяет существенно сокращать время работы алгоритма в случае большой размерности задачи. Вторая модификация основана на возможности метода определять неактивные ограничения задачи. Представлены результаты вычислительных экспериментов с использованием модифицированных версий метода симплексных погружений при решении тестовых задач квадратичной и выпуклой недифференцируемой оптимизации. A simplex embedding method adapted for solving convex optimization problems with a large amount of constraints is considered. Two modifications of the method are proposed for better performance. First of them uses a more economical approach to the residual computation for constraints, which allows one to significantly reduce the execution time of the algorithm in the case of a large amount of constraints. One of the important peculiarities of the simplex embedding method is its ability to find inactive constraints. This property of the method is used as the basis for its second modification. The numerical results obtained when solving a number of quadratic and convex nondifferentiable optimization problems show the efficiency of the proposed modifications.

Trainable Undersampling for Class-Imbalance Learning

Undersampling has been widely used in the class-imbalance learning area. The main deficiency of most existing undersampling methods is that their data sampling strategies are heuristic-based and independent of the used classifier and evaluation metric. Thus, they may discard informative instances for the classifier during the data sampling. In this work, we propose a meta-learning method built on the undersampling to address this issue. The key idea of this method is to parametrize the data sampler and train it to optimize the classification performance over the evaluation metric. We solve the non-differentiable optimization problem for training the data sampler via reinforcement learning. By incorporating evaluation metric optimization into the data sampling process, the proposed method can learn which instance should be discarded for the given classifier and evaluation metric. In addition, as a data level operation, this method can be easily applied to arbitrary evaluation metric and classifier, including non-parametric ones (e.g., C4.5 and KNN). Experimental results on both synthetic and realistic datasets demonstrate the effectiveness of the proposed method.