Auxiliary Problem Research Articles

Adaptive Auxiliary Task Selection for Multitasking-Assisted Constrained Multi-Objective Optimization [Feature

Solving constrained multi-objective optimization problems (CMOPs) is one of the most popular research topics in the multi-objective optimization community. Various approaches based on different algorithmic strategies have been proposed for benchmark CMOPs with different features and challenges. Although most of these algorithms employ one or more fixed strategies, determining the most suitable strategy is challenging and problem-dependent. This work presents a general multitasking framework comprising a main task for the original CMOP and an arbitrary number of auxiliary tasks based on different strategies for the auxiliary problems. Reinforcement learning techniques are utilized according to the population state to choose the most suitable auxiliary task during the evolutionary process. As instantiations, Q-Learning and Deep Q-Learning are successfully applied in the framework. Accordingly, two novel algorithms are developed to solve CMOPs. The results demonstrate the effectiveness of the proposed approaches on five benchmark CMOP test suites and real-world problems compared to 11 advanced methods.

Two-dimensional Inverse Boundary Value Problem for a Third-order Pseudo-hyperbolic Equation with an Additional Integral Condition

In this paper we study an inverse boundary value problem with an unknown timedependent coefficient for a third-order pseudo-hyperbolic equation with an additional integral condition. The definition of the classical solution of the problem is given. The essence of the problem is that it is required together with the solution to determine the unknown coefficient. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, the transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then the transition to the original inverse problem is again made, as a result, a conclusion is made about the solvability of the original inverse problem.

Differential Flatness of Slider–Pusher Systems for Constrained Time Optimal Collision Free Path Planning

Abstract In this work, we show that the differential kinematics of slider–pusher systems are differentially flat assuming quasi-static behavior and frictionless contact. Second, we demonstrate that the state trajectories are invariant to time-differential transformations of the path parametrizing coordinate. For one this property allows to impose arbitrary velocity profiles on the slider without impacting the geometry of the state trajectory. This property implies that certain path planning problems may be decomposed approximately into a strictly geometric path planning and an auxiliary throughput speed optimization problem. Building on these insights, we elaborate a numerical approach tailored to constrained time optimal collision free path planning and apply it to the slider–pusher system.

Integer programming column generation: accelerating branch-and-price using a novel pricing scheme for finding high-quality solutions in set covering, packing, and partitioning problems

AbstractLarge-neighbourhood search (LNS) heuristics are important mathematical programming techniques that search for primal feasible solutions by solving an auxiliary problem with a restricted feasible region. Extending such powerful generic LNS heuristics to the branch-and-price context is inherently challenging. The most prominent challenges arise from the fact that in branch-and-price algorithms, columns are generated with the sole aim to solve linear programming relaxations. Hence, the ability to form integer feasible solutions is not considered during the generation of columns. Without any changes to the standard pricing schemes, the potential of deploying generic LNS heuristics within a branch-and-price procedure is severely limited. This paper proposes a matheuristic, based on an LNS heuristic framework, where the novelty is a customised pricing scheme for generating columns to solve an auxiliary problem. The theoretical foundation for this pricing scheme is a set of optimality conditions for integer programs. From this foundation, a column generation strategy is developed for finding columns that are likely to be of use in high-quality primal feasible solutions for the original problem. The proposed matheuristic is implemented in the generic branch-price-and-cut solver GCG. On a broad test set comprising classical block diagonal structured instances and general instances from the MIPLIB 2017 Collection, the computational results show a significant improvement to the solving performance of GCG.

A Green’s function-based approach to the concentration tensor fields in arbitrary elastic microstructures

Computational homogenization based on FEM models is the gold standard when it comes to homogenization over a representative volume element (RVE), of so-called complex material microstructures, i.e., such which cannot be satisfactorily represented by an assemblage of homogeneous subdomains called phases. As a complement to the aforementioned models, which depend on the boundary conditions applied to the representative volume element and which, as a rule, do not give direct access to the macro-micro-relations in terms of concentration tensors, we here introduce a Green’s function-based homogenization method for arbitrary inhomogeneous microstructures: Inspired by the ideas underlying traditional phase-based homogenization schemes, such as the Mori-Tanaka or the self-consistent model, the new method rests on mapping, through the strain average rule, the microscopic strain fields associated with an auxiliary problem to the macroscopic strains subjected to the RVE. Thereby, the auxiliary problem is defined on a homogeneous infinite matrix subjected to homogeneous auxiliary strains and to inhomogeneous (fluctuating) polarization stresses representing the fluctuations of the microstiffness field, i.e., the complex microstructure within the RVE. The corresponding microscopic strains appear as the solution of a Fredholm integral equation, delivering a multilinear operator linking the homogeneous auxiliary strains to the microscopic strains. This operator, together with the aforementioned mapping, eventually allows for completing the model in terms of concentration tensor and homogenized stiffness quantification. This is illustrated by example of a sinusoidally fluctuating microstructure, whereby the corresponding singular convolution integrals are analytically evaluated from the solution of the Poisson’s equation, and this evaluation strategy is then analytically verified through a Cauchy principal value analysis, and numerically validated by a state-of-the-art FFT homogenization procedure. For the given example, the novel analytical method is several thousand times faster than an FTT-based computational homogenization procedure.

Sensitivity of Seawater Temperature and Salinity to their Restoring Times, Appending in the Boundary Conditions for these Variables at the Free Surface of the Laptev Sea in the No-Ice Period

In order to study sensitivity of seawater temperature and salinity in the no-ice Laptev Sea to their restoring times, appending in the boundary conditions for these variables at the free surface, the 3D finite-element hydrostatic model QUODDY-4 and the indirect means of describing internal tidal waves have been applied. The latter is based on the use of the corrected eddy diffusivity representing the sum of the same uncorrected diffusivity and the diapycnal diffusivity. The first of them characterizes the influence of non-tidal factors, computed using 2.5-level eddy closure scheme, the second, determined by the ratio of internal tidal waves induced baroclinic tidal energy dissipation to the buoyancy frequency in square, the influence of purely tidal factor. The estimate of this dissipation obtained from a solution of auxiliary problem on the internal tidal waves dynamics and energetics is attached. Seawater temperature and salinity in the subsurface and near-bottom layers of the Sea and also their vertical distributions along the 120E meridional transection obtained for the strong, moderate, and mixed restoring times are discussed. As a result, it is clarified that seawater temperature and salinity are weakly sensitive to changes in the restoring time. The saying follows from a comparison of model averaged (over a chosen period and by the Sea area) values of seawater temperature and salinity, and from their local vertical profiles, computed with regard to accepted estimates of the restoring time.

Control Problem Concerned With the Process of Heating a Thin Plate

Previously, a mathematical model for the following problem was considered. On a part of the border of the right rectangle there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some region reaches some given value. In this paper, we consider a boundary control problem associated with a parabolic equation on a right rectangle. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation. In addition, the definition of the generalized solution of the given initialboundary problem is given in the article and the existence of such a solution is proved. The solution of Volterra’s integral equation was found by the Laplace transform method and the existence theorem for admissible control functions was proved. It is also shown that the initial value of the admissible control function is equal to zero using the change of variable in the integral equation. The proof of this comes from the fact that the kernels of the integral equations are positive and finite, and the system has a single-valued solution.Ранее была рассмотрена математическая модель следующей задачи. На части границы правого прямоугольника расположен нагреватель с регулируемой температурой. Требуется найти такой режим его работы, чтобы средняя температура в каком-либо районе достигала некоторого заданного значения. В данной работе рассматривается задача граничного управления, связанная с параболическим уравнением на правом прямоугольнике. На части границы рассматриваемой области указано значение решения с управляющим параметром. Ограничения на управление задаются таким образом, чтобы среднее значение решения в некоторой части рассматриваемой области принимало заданное значение. Вспомогательная задача решается методом разделения переменных, а рассматриваемая задача сводится к интегральному уравнению Вольтерра. Кроме того, в статье дается определение обобщенного решения данной начально-краевой задачи и доказывается существование такого решения. Методом преобразования Лапласа найдено решение интегрального уравнения Вольтерра и доказана теорема существования допустимых управляющих функций. Также показано, что начальное значение допустимой функции управления равно нулю с помощью замены переменной в интегральном уравнении. Доказательство этого исходит из того, что ядра интегральных уравнений положительны и конечны, а система имеет однозначное решение.

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.

Highly reconfigurable silicon integrated microwave photonic filter towards next-generation wireless communication

Integrated microwave photonic filters (IMPFs) are capable of offering unparalleled performances in terms of superb spectral fineness, broadband, and more importantly, the reconfigurability, which encounter the trend of the next-generation wireless communication. However, to achieve high reconfigurability, previous works should adopt complicated system structures and modulation formats, which put great pressure on power consumption and controlment, and, therefore, impede the massive deployment of IMPF. Here, we propose a streamlined architecture for a wideband and highly reconfigurable IMPF on the silicon photonics platform. For various practical filter responses, to avoid complex auxiliary devices and bias drift problems, a phase-modulated flexible sideband cancellation method is employed based on the intensity-consistent single-stage-adjustable cascaded-microring (ICSSA-CM). The IMPF exhibits an operation band extending to millimeter-wave (≥30 GHz), and other extraordinary performances including high spectral resolution of 220 MHz and large rejection ratio of 60 dB are obtained. Moreover, Gb/s-level RF wireless communications are demonstrated for the first time towards real-world scenarios. The proposed IMPF provides broadband flexible spectrum control capabilities, showing great potential in the next-generation wireless communication.

On the universality and membership problems for quantum gates

We study the universality and membership problems for gate sets consisting of a finite number of quantum gates. Our approach relies on the techniques from compact Lie group theory. We also introduce an auxiliary problem called the subgroup universality problem, which helps in solving some instances of the membership problem and can be of interest on its own. The resulting theorems are mainly formulated in terms of centralizers and the adjoint representations of a given set of quantum gates.

Numerical method to solution of generalized model Buckley-Leverett in a class of discontinuous functions

A new numerical method is proposed for solving the generalized Buckley-Leverett problem, which describes the movement of two-phase mixtures of Bazhenov bed sediments in a class of discontinuous functions. To this end, we introduce an auxiliary problem that has advantages over the main problem, and using these advantages, an original finite difference method to solve of the auxiliary problem is developed. Using the suggested auxiliary problem, a solution which expresses exactly all physical characteristics of the problem is obtained.

On the solvability of the initial-boundary value problem for a nonlocal hyperbolic equation

In this paper, we consider a partial differential equation with involutively transformed arguments in a rectangular domain. The considered equation is a non-local analog of the second-order hyperbolic type equation. This equation is subject to initial-boundary conditions, and the order of the boundary operators exceeds the order of the equation. Questions of correctness of the considered problem are investigated. To solve the problem, the Fourier method is used, i.e. separation of variables method. The properties of eigenfunctions and eigenvalues of the corresponding spectral problem are studied. For the main problem under consideration, theorems on the uniqueness and existence of a solution are proved. When proving the theorem on the uniqueness of the solution, the problem under study is reduced to two auxiliary, homogeneous initial-boundary value problems for a classical equation of hyperbolic type. The resulting equations depend on the coefficients of the main equation and certain conditions are imposed on them. Further, using the completeness of the eigenfunctions of the auxiliary spectral problem, the solution of the main problem is sought in the form of a series in this system. For the unknown coefficients of the series, a system of ordinary differential equations with high-order boundary conditions is obtained. Solving these problems, we find an explicit form of the solution of the main problem under study.

Optimal control in the boundary value problem for elliptic equations with degeneration

The problem of optimal control of the system described by the oblique derivative problem forthe elliptic equation of the second order is studied. Cases of internal and boundary managementare considered. The quality criterion is given by the sum of volume and surface integrals.The coefficients of the equation and the boundary condition allow power singularities of arbitraryorder in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special H\"{o}lder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will bethe solution to the given problem.
 The necessary and sufficient conditions for the existence of the optimal solution of the systemdescribed by the boundary value problem for the elliptic equation with degeneracy have been established.

Finding hate speech with auxiliary emotion detection from self-training multi-label learning perspective

Hate Speech Detection (HSD) aims to identify whether a text contains hate speech content, which often refers to discrimination and is even associated with a hate crime. The mainstream methods jointly train the HSD problem with relevant auxiliary problems, e.g., emotion detection and sentiment analysis, under the paradigm of Multi-Task Learning (MTL). In this paper, we improve HSD by integrating it with emotion detection, since we take inspiration from the potential correlations between hate speech and certain negative emotion states, which have been studied theoretically and empirically. To be specific, we can concatenate their hateful labels and predicted emotion states as pseudo-multiple labels for hate speech samples, formulating a pseudo-Multi-Label Learning (MLL) problem. Beyond the existing MTL-HSD methods, we further incorporate this pseudo-MLL problem and solve it by capturing the correlations between hate speech and negative emotion states, so as to improve the performance of HSD. Based on these ideas, we propose a novel HSD method named the Emotion-correlated Hate Speech DetectOR (EHSor). We conduct extensive experiments to evaluate EHSor, and the results show that it can consistently outperform the existing HSD methods across benchmark datasets.

A First-Order Primal-Dual Method for Nonconvex Constrained Optimization Based on the Augmented Lagrangian

Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics, and data analytics. In this paper, based on the augmented Lagrangian function, we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of [Formula: see text], where k is the number of iterations. Moreover, under an additional error bound condition (to be called HVP-EB in the paper) with exponent [Formula: see text], we further show the global convergence of NAPP-AL. Additionally, if [Formula: see text], then we furthermore show that the convergence rate is in fact linear. Finally, we show that the well-known Kurdyka-Łojasiewicz property and the Hölderian metric subregularity imply the aforementioned HVP-EB condition. We demonstrate that under mild conditions, NAPP-AL can also be interpreted as a variant of the forward-backward operator splitting method in this context. Funding: This work was supported by the National Natural Science Foundation of China [Grant 71871140].

Optimal Vaccination Policies for COVID-19 Considering Vaccine Doses Delays

This work introduces a methodology to determine an optimal vaccination strategy for the COVID-19 disease with limited vaccination capacity, considering the delays between vaccine doses in order to minimize the number of deaths. The evolution of the pandemic in the city of Tandil, province of Buenos Aires, Argentina, is studied through a compartmental model, where the parameters involved are calibrated according to official data. An auxiliary age-structured optimal control problem is introduced, where optimal policies are obtained taking into account the characteristics of the vaccine and the vaccination capacity. Through numerical simulations, the policy obtained is compared with other feasible strategies, considering or not strict isolation measures.

Computational Lower Bounds of the Maxwell Eigenvalues

A method to compute guaranteed lower bounds to the eigenvalues of the Maxwell system in two or three space dimensions is proposed as a generalization of the method of Liu and Oishi [SIAM J. Numer. Anal., 51 (2013), pp. 1634–1654] for the Laplace operator. The main tool is the computation of an explicit upper bound to the error of the Galerkin projection. The error is split into two parts. One part is controlled by a hypercircle principle and an auxiliary eigenvalue problem. The second part requires a perturbation argument for the right-hand side replaced by a suitable piecewise polynomial. The latter error is controlled through the use of the commuting quasi-interpolation by Falk and Winther and computational bounds on its stability constant. This situation is different from the Laplace operator where such a perturbation is easily controlled through local Poincaré inequalities. The practical viability of the approach is demonstrated in test cases for two and three space dimensions.

Robust MPC for polytopic uncertain systems via a high-rate network with the round-robin scheduling.

This article is concerned with the robust model predictive control (RMPC) problem for polytopic uncertain systems under the round-robin (RR) scheduling in the high-rate communication channel. From a set of sensors to the controller, several sensors transmit the data to the remote controller via a shared high-rate communication network, data collision might happen if these sensors start transmissions at the same time. For the sake of preventing data collision in the high-rate communication channel, a communication scheduling known as RR is used to arrange the data transmission order, where only one node with token is allowed to send data at each transmission instant. In accordance with the token-dependent Lyapunov-like approach, the aim of the problem addressed is to design a set of controllers in the framework of RMPC such that the asymptotical stability of the closed-loop system is guaranteed. By taking the effect of the underlying RR scheduling in the high-rate communication channel into consideration, sufficient conditions are obtained by solving a terminal constraint set of an auxiliary optimization problem. In addition, an algorithm including both off-line and online parts is provided to find a sub-optimal solution. Finally, two simulation examples are used to demonstrate the usefulness and effectiveness of the proposed RMPC strategy.

On the multi-period combined maintenance and routing optimisation problem

This paper focuses on the combined maintenance and routing optimisation problem in a multi-period environment that consists of scheduling maintenance operations for a set of geographically distributed machines, subject to non-deterministic failures with a set of technicians that perform preventive maintenance and corrective operations. This study uses a two-step approach based on column generation. The first step consists of a maintenance model that determines the optimal time until the next preventive maintenance operation, and each machine’s maintenance frequency while minimising the total expected maintenance costs. The second step involves a routing model that assigns and schedules maintenance operations for each technician over the planning horizon while minimising the routing and maintenance costs. We formulate the second step problem as a periodic vehicle routing problem and to solve it, we propose a column generation approach. The master problem is a set partitioning formulation that indicates which machines must be visited in the planning horizon. This decomposition has two auxiliary problems: the first one is an elementary shortest-path problem to find a feasible path for each period and the second one finds a set of visiting periods for each machine. Our approach balances the maintenance cost, routing cost, and failure probabilities.

On a problem for a delay differential equation

The article considers a nonlinear boundary value problem for a linear delay differential equation. To solve the problem, the idea of parametrization method, namely, the interval at which the problem is being considered, is divided into subintervals whose lengths do not exceed the values of the constant delay; constant parameters are introduced at the left ends of these intervals; a new unknown function is introduced at each subinterval. Thus, the problem under consideration is reduced to an equivalent multipoint boundary value problem for differential equations with delay containing parameters. Auxiliary Cauchy problems without delay with zero initial conditions at the left ends of the subintervals are consistently considered on each of the subintervals. Using an analog of the Cauchy formula to represent the solution of a system of linear differential equations on each of the subintervals and given nonlinear boundary conditions, an algebraic system with respect to unknown parameters is obtained. The article proposes an algorithm for finding a solution to a multipoint boundary value problem for differential equations with a delay containing parameters. At each step of the algorithm, a system of nonlinear algebraic equations is solved to determine the values of the parameters, and an analog of the Cauchy formula is used to obtain solutions to auxiliary Cauchy problems. The results obtained are demonstrated on a test problem.