Dual Formulation Research Articles

A Filter-SQP strategy for training Support Vector Machine models

This paper introduces a filtering strategy for addressing optimization problems arising in binary Support Vector Machine classification. The training optimization problem aims to solve the dual formulation which involves a quadratic objective function subjected to a linear and box constraints. Our approach employs a Filter algorithm with Sequential Quadratic Programming iterations that minimize the quadratic Lagrangian approximations. Notably, we utilize the exact Hessian matrix in our numerical experiments to seek the desired classification function. Moreover, we present a Filter algorithm combined with the Augmented Lagrangian method aiming to accelerate the algorithm convergence. To substantiate our method's effectiveness, we conduct numerical experiments through MATLAB, comparing outcomes with alternative methodologies detailed in existing literature. Numerical experiments shows that the Filter--SQP combined with Augmented Lagrangian method is competitive and efficient method compared with an interior-point based solver and LIBSVM software in relation of classification metrics and CPU-time.

Dual full-wave formulations with Radiant Electric Circuit Element boundary conditions — Application to monopole antenna modeling

Electric Circuit Element (ECE) are special boundary conditions that enable a natural coupling between electromagnetic devices and circuits. ECE can be generalized to allow for parts on the domain boundary where absorbing boundary conditions can be set, to model the unbounded space. This paper illustrates the implementation of radiant ECE in the 2D and 3D-finite element method, for a full-wave time-harmonic field in dual frequency-domain E and H formulations. The weak formulations are implemented in the free environment Open Numerical Engineering LABoratory (onelab). The formulations are proven to be well-posed (existence, uniqueness and stability of the weak solution are proven) and validated for a monopole antenna.

A dual approach to agency problems

This paper presents a dual approach to the standard model of moral hazard. We formulate the dual of the principal–agent problem under the assumption that the incentive constraint can be replaced by a local constraint (the first-order approach), to examine whether the relaxed agency problem yields a candidate solution. The dual formulation generates a convex conjugate, which transforms the agent’s utility from compensation into a dual functional. The dual problem features a simple convex structure, which enables us to perform a comprehensive analysis for the agency problem. We derive novel and more tractable conditions for existence and uniqueness of a solution to the problem with the dual elements. Furthermore, the approach to the dual problem provides illuminating insights into the previous nonexistence results.

Energy efficiency lags and welfare boons: Understanding the rebound and welfare effects through China's urban households

Improving efficiency is a core measure to reduce energy demand and associated emissions, yet actual performance invariably lags expectations. This is widely understood to occur due to the rebound effect, where the cost savings from improving efficiency are spent on other energy and emissions-generating activities. Notwithstanding the environmental negative of rebound, its associated potential welfare benefit is less understood, thus complicating the analysis of sustainable development outcomes. To develop a deeper understanding of rebound and associated welfare effects, this study innovates by considering both in a unified framework using a hybrid methodology and Taylor expansion. The consumer welfare effect is specified in dual form to the rebound effect, and a classification of four consumer types is obtained accordingly. The methodology is applied to China's urban households from 1995 to 2018. The rebound effects for carbon dioxide, sulfur dioxide, nitrogen oxides, and particulates emissions, are found to be in the range of 77–80 %. The welfare effect, a potentially desirable outcome, is estimated to be 4 %. The results demonstrate that rebound has drained most of the expected environmental improvement, while simultaneously generating a substantial consumer welfare benefit associated with energy consumption. It is concluded that the average urban resident in China can be appropriately characterised as ‘insufficient in demand’, implying a challenge to short-run transition. The conceptual and analytical novelty of the study can support the proposal of innovative policy recommendations, through re-aligned energy efficiency policies, that can minimise rebound, while improving welfare in parallel.

DESIGNING AN EFFECTIVE EDUCATIONAL PRODUCTION ENVIRONMENT IN VOCATIONAL SCHOOLS UNDER DUAL VOCATIONAL EDUCATION

Relevance: The issue of introducing dual education, in particular the design of a quality educational and production environment of a professional education institution under such conditions, is becoming more and more relevant. At the same time, it is important to take into account both objective and subjective factors affecting the specified process, as well as to determine the directions and prospects for the development of a high-quality educational and production environment of a professional education institution. The article aims to discover how to design an effective educational production environment in vocational schools in the conditions of dual vocational education, create it in the schools involved in the experiment, assess its quality under appropriate parameters and provide relevant recommendations. Methods: analysis of scientific and pedagogical sources and regulatory framework; study of the work practice of vocational education institutions according to research issues; SWOT analysis of the introduction of elements of a dual form of education in the professional training of future specialists, surveys of vocational education applicants, pedagogical workers of vocational education institutions and representatives of enterprises; modeling, analysis, synthesis and generalization of scientific material for the development and implementation of methodological tools; systematization and generalization of research results. The results. As part of this research, an effective educational production environment in the conditions of dual vocational education was designed in vocational schools involved in the experiment after introducing the authors’ methodical tools. The quality of the designed environment was assessed at the beginning and at the end of the formative experiment (the training course for vocational education teachers was organized; the authors’ methodical tools were implemented) with the help of such parameters as modality, extension, intensity, awareness, generalization, dominance, social activity, mobility. A detailed analysis of the obtained results shows a significant increase in the quality of the specified parameters of an educational production environment in vocational schools with some elements of dual vocational education at a high level. Conclusions. It is essential to support various innovations and initiatives of all the actors in the educational process to strengthen a creative atmosphere in vocational schools and, thus, prepare creative specialists who should act as proactive and free citizens of their country. It is crucial to build a constructive dialogue between all the actors in dual vocational education based on the principles of partnership and consensus to boost their motivation for joint productive activity and responsibility for its results and make their actions more coordinated. It is important to regularly update technological equipment in vocational schools and enterprises providing their production sites for practical training and promote continuing professional development of vocational education teachers and mentors at the enterprises to ensure high-quality vocational training of future specialists in the conditions of dual vocational education. Finally, it is advisable to improve educational programmes and curricula following the needs of enterprises (employers), as well as the fast-moving requirements of the labour market, to support the flexibility of the educational process.

Exploring inelasticity in the S-matrix Bootstrap

The modern S-Matrix Bootstrap provides non-perturbative bounds on low-energy aspects of scattering amplitudes, leveraging the constraints of unitarity, analyticity and crossing. Typically, the solutions saturating such bounds also saturate the unitarity constraint as much as possible, meaning that they are almost exclusively elastic. This is expected to be unphysical in d>2 because of Aks' theorem. We explore this issue by adding inelasticity as an additional input, both using a primal approach in general dimensions which extends the usual ansatz, and establishing a dual formulation in the 2d case. We then measure the effects on the low-energy observables where we observe stronger bounds than in the standard setup.

Optical solitons of the complex Ginzburg-Landau equation having dual power nonlinear form using $\varphi^{6}$-model expansion approach

This paper employs a novel $\varphi ^{6}$-model expansion approach to get dark, bright, periodic, dark-bright, and singular soliton solutions to the complex Ginzburg-Landau equation with dual power-law non-linearity. The dual-power law found in photovoltaic materials is used to explain nonlinearity in the refractive index. The results of this paper may assist in comprehending some of the physical effects of various nonlinear physics models. For example, the hyperbolic sine arises in the calculation of the Roche limit and the gravitational potential of a cylinder, the hyperbolic tangent arises in the calculation of the magnetic moment and the rapidity of special relativity, and the hyperbolic cotangent arises in the Langevin function for magnetic polarization. Frequency values, one of the soliton's internal dynamics, are used to examine the behavior of the traveling wave. Finally, some of the obtained solitons' three-, two-dimensional, and contour graphs are plotted.

Shallow water memory: Stokes and Darwin drifts

It has been shown in [SciPost Phys. 14, 102 (2023)] that shallow water in the Euler description admits a dual gauge theory formulation. We show in the Lagrange description this gauge symmetry is a manifestation of the 2 dimensional area-preserving diffeomorphisms. We find surface charges associated with the gauge symmetry and their algebra, and study their physics in the shallow water system. In particular, we provide a reinterpretation of the Kelvin circulation theorem in terms of conserved charges. In the linear shallow water case, the charges form a u(1) current algebra with level proportional to the Coriolis parameter over the height of the fluid. We also study memory effect for the gauge theory description of the linearized shallow water and show Euler, Stokes and Darwin drifts can be understood as a memory effect and/or change of the surface charges in the gauge theory description.

A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle

A Primal-Worst-Equals-Dual-Best Perspective on Robust and Distributionally Robust Optimization In the paper “A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle,” Jianzhe Zhen, Daniel Kuhn, and Wolfram Wiesemann develop a generalized “primal-worst-equals-dual-best” principle that establishes strong duality between semi-infinite primal worst and nonconvex dual best formulations of robust and distributionally robust nonlinear optimization problems. Their theory offers an alternative characterization of (distributionally) robust optimization problems that bypasses the need to mobilize the machinery of abstract semi-infinite duality theory. The paper will be of interest to researchers and practitioners in the field of optimization under uncertainty.

Metasurface characterization based on eigenmode analysis and averaging of electromagnetic fields

A fully numerical homogenization technique for the retrieval of the effective surface susceptibilities of a periodic composite metasurface is developed in this work. We utilize the so-called dual field-flux finite element formulation scheme to accurately calculate the eigenmodes of a composite periodic metasurface, a scheme that possesses a crucial advantage: The capability of evaluating all field components and their derivatives accurately and with the same order of approximation is a requirement for our proposed technique. Next, we derive the generalized sheet transition condition equations for a general bianisotropic metasurface, which correlate the field components on either sides of the metasurface with its surface susceptibilities. At this point, we establish a new field averaging scheme for the acquisition of the average field components of the modes supported by the metasurface. Combining this computational information, we derive a set of linear algebraic equations based on the GSTCs and the average electromagnetic fields and numerically solve it, so as to obtain the effective surface susceptibilities of a bianisotropic metasurface. Comparison with the results of other techniques in the literature shows very good agreement, relatively to the resonance behavior of the returned values and their position at the frequency spectrum. The advantages that distinguish the proposed technique over other related methods are its foundation on the intrinsic modal information of the eigenmodes supported by the metasurface and its independence of any wave excitation schemes or involvement of analytical polarizability calculations.

A necessary condition for extremality of solutions to autonomous obstacle problems with general growth

Let us consider the autonomous obstacle problem minv∫ΩF(Dv(x))dxon a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity. Our aim is to find a necessary condition for the extremality of the solution, which exists and it is unique, thanks to a primal–dual formulation of the problem. The proof is based on classical arguments of Convex Analysis and on Calculus of Variations’ techniques.

Hiemenz flow for a micropolar nanofluid with bidirectional flexible surface and heat transfer

The steady Hiemenz stagnation-point flow of a flexible membrane moving bidirectionally, for a micropolar copper–water nanofluid model is studied. The flow field is a unison of wall transpiration, velocity slip and stagnation-point flow for a 2D case, generating unique or dual form of exact similarity solutions. The results reveal that a unique solution exists for the stretching sheet, depending entirely on a given set of parameters, while two solutions fall within the physically recognizable region for the case of a shrinking sheet. A set of new exact solutions characterized by the stretching/shrinking strength parameter d are also found with d considering different values. It is found that the numerical integration for skin friction, couple stress friction and temperature gradient leads to twofold solutions.

An accelerated primal‐dual method for semi‐definite programming relaxation of optimal power flow

AbstractThe application of a semi‐definite programming (SDP) approach to the Alternating Current Optimal Power Flow problem has attracted significant attention in recent years. However, the SDP relaxation of optimal power flow (OPF) can be computationally intensive and lead to memory issues when dealing with large‐scale power systems. To overcome these challenges, we have developed APD–SDP, an optimisation solver based on a first‐order primal–dual algorithm. This framework incorporates various acceleration techniques, such as rescaling, step size decay and reset, adaptive line search, and restart, to improve efficiency. To further speed up computations, we have developed a customised eigenvalue decomposition component by exploiting the 3 × 3 block structure in the dual SDP formulation. Experimental results demonstrate that APD–SDP outperforms other commercial and open‐source SDP solvers on large‐scale and high‐dimensional PGLib‐OPF datasets.

Weak constraint 4D-Var data assimilation in the Regional Ocean Modeling System (ROMS) using a saddle-point algorithm: Application to the California Current Circulation

The saddle-point formulation of weak constraint 4-dimensional variational (4D-Var) data assimilation has been developed for the Regional Ocean Modeling System (ROMS), and is applied here to the California Current System (CCS). Unlike the conventional primal and dual forcing formulation of weak constraint 4D-Var, the saddle-point formulation can be efficiently parallelized in time, leading to a substantial decrease in wall-clock run time. The performance of the ROMS saddle-point 4D-Var algorithm is assessed here and compared to that of the dual forcing formulation which is the current standard in ROMS. While the rate of convergence of the saddle-point formulation is slower than the dual forcing formulation, the increase in computational speed due to time-parallelization can more than compensate for the additional inner-loop iterations required by the saddle-point algorithm in the CCS configuration considered here. Further gains in performance can be achieved by running the 4D-Var inner-loop iterations at reduced resolution and/or reduced arithmetic precision. The results presented here indicate that in high performance computing environments with large numbers of available compute cores, the saddle-point formulation of 4D-Var could significantly reduce the run-time compared to the dual forcing formulation for large data assimilation problems.

Dual form of education is the basis of the integration of students into the modern labor market

The dual form of education is a critical component of integrating young people into the modern labor market, as it helps reduce the unemployment rate and ensures the training of highly qualified specialists. The main goal of implementing elements of a dual form of education in Ukraine is to eliminate the main shortcomings of traditional forms and methods of training future qualified specialists, namely, to overcome the gap between theory and practice, education, and production, and to promote the improvement of the quality of personnel training, considering the requirements of employers by the demand of the modern labor market. Therefore, the opportunity to simultaneously acquire both theoretical knowledge and practical experience at real workplaces gives higher education graduates a competitive advantage in the labor market. The article aims to study modern trends in the implementation of dual education as a basis for integrating students into the modern labor market. According to official data, over the past few years, there has been a steady trend towards an increase in the number of unemployed youths, which is due to both unresolved youth employment problems and the spread of passive, unregulated, destructive behavior patterns of the young generation in the conditions of the modern domestic labor market, which, unfortunately, is negative signs of its functioning. Post-war recovery and development of all spheres of the domestic economy requires. It was determined that the use of the potential of various forms of higher education, among which the dual form of education occupies an important place. In the forced conditions of distance learning, it is challenging to establish practical training for higher education students and quality control over their acquisition of the necessary competencies. Therefore, the use of the means of organizing dual education will contribute to acquiring the necessary professional knowledge and professional competencies that meet the requirements of employers and the labor market. Attracting qualified specialists to work is a strategic investment in the social socio-economic development of the state. Keywords: dual form of acquiring professional competences, educational process, higher education, competence approach, participants in the educational process, university stakeholders.

Multi-constrained optimal reinsurance model from the duality perspectives

In the presence of multiple constraints such as the risk tolerance constraint and the budget constraint, many extensively studied (Pareto-)optimal reinsurance problems based on general distortion risk measures become technically challenging and have only been solved using ad hoc methods for certain special cases. In this paper, we extend the method developed in Lo (2017a) by proposing a generalized Neyman-Pearson framework to identify the optimal forms of the solutions. We then develop a dual formulation and show that the infinite-dimensional constrained optimization problems can be reduced to finite-dimensional unconstrained ones. With the support of the Nelder-Mead algorithm, we are able to obtain optimal solutions efficiently. We illustrate the versatility of our approach by working out several detailed numerical examples, many of which in the literature were only partially resolved.

Guaranteed lower bounds to effective stiffness

AbstractWe present a numerical scheme for obtaining guaranteed (reliable) and arbitrarily close two sided bounds to effective (homogenized) parameters of the linear elasticity problem. For the upper bounds, we use standard finite element (FE) discretization of the so‐called primal problem with preconditioning based on the fast discrete Fourier transformation (FFT). For the lower bounds, we use the dual formulation and some smoother FE approximation spaces. Moreover, instead of solving the discretized dual problem, we can only compute an L2‐orthogonal projection of an auxiliary field built from the primal solution. The projection can be computed easily by FFT and provides a lower bound of almost the same quality as that obtained as the exact solution of the discretized dual problem. In addition, a simple low‐dimensional optimization improves the projected solution. Numerical examples are presented to support the theoretical developments.

Deterministic bridge regression for compressive classification

Pattern classification using a compact representation is a crucial component of machine intelligence. Specifically, it is essential to learn a model with well-regulated parameters to achieve good generalization. Bridge regression provides a mechanism for regulating parameters through a penalized ℓp-norm. However, due to the nonlinear nature of the formulation, an iterative numerical search is typically used to solve the optimization problem. In this work, we propose an analytic solution for bridge regression based on solving a penalized error formulation using an approximated ℓp-norm. The solution is presented in primal form for over-determined systems and in dual form for under-determined systems. The primal form is suitable for low-dimensional problems with a large number of data samples, while the dual form is suitable for high-dimensional problems with a small number of data samples. We also extend the solution to problems with multiple classification outputs. Numerical studies using simulated and real-world data demonstrate the effectiveness of our proposed solution.

A stable variant of linex loss SVM for handling noise with reduced hyperparameters

Support Vector Machine (SVM) primarily uses the hinge loss function with maximum margin. The boundary instances determine the separating hyperplanes. However, in real-world scenarios, noisy instances around the decision boundary can degrade the performance of hinge SVM. To address this issue, researchers introduced the pinball SVM, which uses all the instances to determine the separating hyperplane and has a discontinuous piecewise linear loss function. A more recent variant, the linex loss SVM with NAG, employs a combination of linear and exponential loss functions, resulting in an asymmetric and continuous loss function. Due to the employment of Nesterov accelerated gradient (NAG), the older linex SVM formulation was unstable. This study presents a novel noise-robust variant of linex SVM, termed linexSVM, which is stable in performance compared to the linex SVM with NAG. Additionally, the proposed linexSVM has fewer hyperparameters in the dual formulation than other noise-robust SVM variants, such as pinball SVM, ϵ-insensitive pinball SVM, and ϵ1,ϵ2 pinball SVM. We evaluated the performance of the proposed linexSVM, hinge SVM, and pinball SVM using the UCI dataset repository. The experimental results indicate that the proposed linexSVM outperforms hinge and pinball loss SVMs on datasets with and without noise.

Searching for a Feedback Vertex Set with the link-cut tree

In this paper we study the Feedback Vertex Set problem of an undirected graph. This problem has several practical applications and is know to be NP-Complete. We consider this problem from its dual formulation which consists in finding a largest subset of vertexes whose induced subgraph is a forest, i.e., contains no cycles. Our main goal is to explore how the link-cut tree data structure can be used in this context. This data structure makes it possible to identify and resolve cycles quickly. This property is useful for this problem and allows us to support a simple simulated annealing algorithm. We used the PACE 2016 dataset to test the performance of the resulting algorithm. The worst approximation ratio obtained was about 1.21. The algorithm is very fast. The longest execution time was 103 seconds. To justify the importance of escaping local minimums we also present a simple greedy algorithm.