Standard Second-order Cone Programming Research Articles

Risk-averse distributionally robust optimization for construction waste reverse logistics with a joint chance constraint

The uncertainty of the amount of construction and demolition waste (CDW) generation affects the CDW reverse logistics network service level. We investigate a CDW reverse logistics network location-routing problem considering uncertainties. To minimize the total social cost, a two-stage risk-averse distributionally robust optimization model is developed, which aims to optimize the location and number of CDW disposal facilities and the CDW transportation scheme. We introduce the mean-conditional value at risk measure and a joint chance constraint into our model to consider the government’s risk aversion. The above model is approximated as a standard second-order cone programming model (SOCP) considering a special case. To exactly solve the SOCP, we design an outer approximation algorithm. Several performance tests and a case study are proposed, and sensitivity analysis provides helpful managerial insights.

A convex cone programming based implicit material point method

For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To circumvent these limitations, this paper innovatively proposes a convex cone programming-based implicit MPM (CP-MPM) algorithm, which ensures excellent convergence of solving complex problems involving large deformation, regardless of the chosen time step. In the proposed CP-MPM, the governing equations are initially transformed into a stationary point of a multivariable functional, leveraging the generalized Hellinger-Reissner (HR) variational principle. This stationary point problem is subsequently reformulated as a min-max convex cone optimization problem, with constraints originating from elastoplastic constitutive equations. In addition, a novel particle-based adhesive-frictional contact algorithm is proposed to effectively tackle the interaction between MPM domain and rigid bodies. The contact inequality between material points and rigid bodies is transformed into convex cone constraints, which rigorously prevent material point penetration and facilitate the imposition of irregular boundary conditions. Both elastic and elastoplastic problems involving contact under static or dynamic loading are ultimately represented as standard second-order cone programming (SOCP) problems, which is effectively solved by employing the Primal-Dual Interior Point (PDIP) method. The robustness, accuracy and convergence of the proposed method are validated through a series of elastic and elastoplastic benchmark problems. All results demonstrate the CP-MPM is a very promising method for implicitly solving complex practices.

A stable implicit nodal integration-based particle finite element method (N-PFEM) for modelling saturated soil dynamics

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

Filter design and power allocation in multi-user universal filtered multi-carrier (MU-UFMC) systems

Universal Filtered Multi-Carrier (UFMC) system has been introduced as a candidate for the new generation. UFMC is a multi-carrier system, which can overcome challenges in wireless communications. In fact, the advantages of multi-carrier systems over single-carrier systems include higher bandwidth gain and higher robustness against frequency-selective fading. Generally, most multi-carrier systems suffer from notoriously sensitive to symbol-time offset (STO) and carrier-frequency offset (CFO). However, since UFMC systems utilize subband filters, sensitivity to CFO and STO can be reduced by designing efficient subband filters. In this paper, filter design problem is addressed in order to mitigate interference caused by effect of CFO in a multi-user UFMC (MU-UFMC) system under fair optimization framework. Since filter design problem is formulated as a non-convex optimization problem, we present two different approach in order to solve it. In the first approach, the formulated problem is relaxed so that one is converted to a standard second-order-cone programming (SOCP). In the second approach, we present a proposed algorithm with low complexity based on some lemma and relaxation. Then power allocation problem is solved to maximize the minimum achievable rate among all the users in a MU-UFMC system. The simulation results show the proposed schemes for filter design and power allocation outperform the benchmark schemes.

Nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic modelling of saturated soils under large deformation

This paper presents the nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic analysis of saturated soils subject to large deformation, utilising the generalised Hellinger-Reissner variational principle to reformulate the governing equations for saturated soil dynamics into a min–max optimisation problem. With finite element discretisation and nodal integration over cells, the problem is transformed into a standard second-order cone programming problem, efficiently resolved using the advanced primal–dual interior point method. The N-PFEM method has several advantages, including the use of linear triangular elements without volumetric locking issues, the avoidance of regularisation techniques, and the elimination of tedious variable mapping after remeshing. The numerical model is validated for large deformation analysis of saturated soils with a series of benchmarks against available analytical and numerical solutions, with a case study of the deformation of an embankment considering stone column reinforcement also carried out. This N-PFEM framework offers an effective and efficient simulation approach for the evolutionary behaviour of saturated soils with large deformation in complex geotechnical configurations of practical relevance.

A novel centroid-enriched edge-based smoothed radial point interpolation method for upper bound limit analysis

This paper presents a novel centroid-enriched edge-based smoothed radial point interpolation method (CE-ESRPIM) to overcome the volumetric locking problem and increase the numerical accuracy for upper bound limit analysis. Based on the edge-based smoothed radial point interpolation method (ESRPIM), the centroid of the triangle element is used as a field node to discretize the problem domain. Simultaneously, the local support node selection scheme, which is established based on a modified centroid-enriched triangle mesh (CE-TScheme), is proposed to realize local radial point interpolation (RPIM). Using the Mohr–Coulomb yield criterion and associated flow rule, the upper bound theorem in the plane strain condition is discretized into a standard second-order cone programming (SOCP) problem through the CE-ESRPIM, which is solved using the primal–dual interior-point algorithm (PDIP). The proposed approach with the enriched centroid considered as the field node is straightforward and efficient and can directly inherit the formulation framework of the traditional ESRPIM. Moreover, this method has been proven to be effective in eliminating the volumetric locking problem under incompressible conditions.

A generalized Hellinger-Reissner variational principle and its PFEM formulation for dynamic analysis of saturated porous media

In this paper, a novel mathematical programming formulation is derived based on the u-p form for the dynamic analysis of saturated porous media. A mixed finite element is used for the interpolation of field variables and after discretization the formulation is remolded into a standard second-order cone programming problem that can be resolved using modern optimization engines. The proposed optimization-based computational scheme is verified against typical benchmarks such as the dynamic consolidation problem and the wave propagation in saturated soils. To tackle issues such as mesh distortions and severe free-surface evolutions resulting from large deformations, the scheme is further implemented into the PFEM framework. The capability of the proposed method for analyzing porous media with large deformations is illustrated by modelling the collapse of a saturated granular column in air and the post-failure process of an embankment due to seepage with results compared to the ones from lab tests and numerical simulations using other approaches such as the material point method and the smoothed particle hydrodynamics method.

Three-dimensional spherical discontinuous deformation analysis using second-order cone programming

In this paper, a new formulation of three-dimensional spherical discontinuous deformation analysis (DDA) based on second-order cone programming has been proposed. Artificial springs with open-close iteration used in classic DDA have been removed, given that improper stiffness parameters might cause numerical problems. Furthermore, to account for irregular granular shapes, a rolling resistance model is incorporated in the variational formulation. The proposed formulation can be cast into a standard second-order cone programming program, which can be solved using efficient off-the-shelf optimisation solvers. The proposed approach is validated by a series of numerical examples.

Sparse wavelet estimation in quantile regression with multiple functional predictors

To study quantile regression in partial functional linear model where response is scalar and predictors include both scalars and multiple functions, wavelet basis are usually adopted to better approximate functional slopes while effectively detect local features. The sparse group lasso penalty is imposed to select important functional predictors while capture shared information among them. The estimation problem can be reformulated into a standard second-order cone program and then solved by an interior point method. A novel algorithm is proposed by using alternating direction method of multipliers (ADMM) which was recently employed by many researchers in solving penalized quantile regression problems. The asymptotic properties such as the convergence rate and prediction error bound have been established. Simulations and a real data from ADHD-200 fMRI data are investigated to show the superiority of our proposed method.

Distributionally robust optimization of an emergency medical service station location and sizing problem with joint chance constraints

An effective Emergency Medical Service (EMS) system can provide medical relief supplies for common emergencies (fire, accident, etc.) or large-scale disasters (earthquake, tsunami, bioterrorism attack, explosion, etc.) and decrease morbidity and mortality dramatically. This paper proposes a distributionally robust model for optimizing the location, number of ambulances and demand assignment in an EMS system by minimizing the expected total cost. The model guarantees that the probability of satisfying the maximum concurrent demand in the whole system is larger than a predetermined reliability level by introducing joint chance constraints and characterizes the expected total cost by moment uncertainty based on a data-driven approach. The model is approximated as a parametric second-order conic representable program. Furthermore, a special case of the model is considered and converted into a standard second-order cone program, which can be efficiently solved with a proposed outer approximation algorithm. Extensive numerical experiments are conducted to illustrate the benefit of the proposed approach. Moreover, a dataset from a real application is also used to demonstrate the application of the data-driven approach.

Upper bound limit analysis using the weak form quadrature element method

• The weak form quadrature element method is applied for upper bound limit analysis. • Gauss–Lobatto quadrature is employed to numerically integrate the integrals in the dual formulations. • The efficiency of the existing numerical limit algorithms is improved. • The well-known volumetric locking occurred for incompressible materials is overcome. Limit analysis is a useful tool for design and safety assessment of structures in civil and geotechnical engineering. In the present study, a newly developed high order algorithm-the weak form quadrature element method is reformulated for upper bound limit analysis. The dual formulations of the kinematic theorem are employed with the nodal stresses chosen as the optimization variables. The weak form equilibrium constraint is numerically integrated by Lobatto integration and then the nodal derivatives are approximated by differential quadrature analogue. The resulting optimization problem is formulated as a standard second-order cone programming problem and solved by the optimization toolbox Mosek. This paper aims to improve the efficiency of the existing numerical limit algorithms especially for problems with singularities such as cracked structures and to overcome the well-known volumetric locking occurred for incompressible materials. Some numerical tests are given to show the accuracy and efficiency of the present method.

Discrete modelling jointed rock slopes using mathematical programming methods

In this paper, a mathematical programming based discontinuum approach is developed for modelling jointed rock slopes. The formulation naturally leads to a standard second-order cone program, which can be solved using efficient optimisation solvers, and a purely static method is derived that does not require artificial damping parameters. Notably, the approach provides somewhat of a unification of two distinct discontinuum approaches: the soft-particle model and the hard-particle model. A series of numerical examples are conducted to validate the proposed approach. The soft-particle model is more versatile than the hard-particle model while the hard-particle model is more efficient.

A computational homogenization approach for limit analysis of heterogeneous materials

The macroscopic strength domain and failure mode of heterogeneous or composite materials can be quantitatively determined by solving an auxiliary limit analysis problem formulated on a periodic representative volume element. In this paper, a novel numerical procedure based on kinematic theorem and homogenization theory for limit analysis of periodic composites is developed. The total velocity fields, instead of fluctuating (periodic) velocity, at microscopic level are approximated by finite elements, ensuring that the resulting optimization problem is similar to the limit analysis problem formulated for structures, except for additional constraints, which are periodic boundary conditions and averaging relations. The optimization problem is then formulated in the form of a standard second-order cone programming problem to be solved by highly efficient solvers. Effects of loading condition, representative volume element architecture, and fiber/air void volume fraction on the macroscopic strength of perforated and fiber reinforced composites are studied in numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.

Second-order cone programming formulation for consolidation analysis of saturated porous media

In this paper, the incremental problem for consolidation analysis of elastoplastic saturated porous media is formulated and solved using second-order cone programming. This is achieved by the application of the Hellinger-Reissner variational theorem, which casts the governing equations of Biot's consolidation theory as a min---max optimisation problem. The min---max problem is then discretised using the finite element method and converted into a standard second-order cone programming problem that can be solved efficiently using modern optimisation algorithms (such as the primal-dual interior-point method). The proposed computational formulation is verified against a number of benchmark examples and also applied to simulate the construction of a road embankment on soft clay.

Linear Precoding Designs for Amplify-and-Forward Multiuser Two-Way Relay Systems

Two-way relaying can improve spectral efficiency in two-user cooperative communications. It also has great potential in multiuser systems. A major problem of designing a multiuser two-way relay system (MU-TWRS) is transceiver or precoding design to suppress co-channel interference. This paper aims to study linear precoding designs for a cellular MU-TWRS where a multi-antenna base station (BS) conducts bi-directional communications with multiple mobile stations (MSs) via a multi-antenna relay station (RS) with amplify-and-forward relay strategy. The design goal is to optimize uplink performance, including total mean-square error (Total-MSE) and sum rate, while maintaining individual signal-to-interference-plus-noise ratio (SINR) requirement for downlink signals. We show that the BS precoding design with the RS precoder fixed can be converted to a standard second order cone programming (SOCP) and the optimal solution is obtained efficiently. The RS precoding design with the BS precoder fixed, on the other hand, is non-convex and we present an iterative algorithm to find a local optimal solution. Then, the joint BS-RS precoding is obtained by solving the BS precoding and the RS precoding alternately. Comprehensive simulation is conducted to demonstrate the effectiveness of the proposed precoding designs.

Granular contact dynamics with particle elasticity

A granular contact dynamics formulation for elastically deformable particles is detailed. The resulting scheme bears some similarity to traditional molecular dynamics schemes in that the consideration of a finite elastic contact stiffness implies the possibility for inter-particle penetration. However, in contrast to traditional molecular dynamics schemes, there are no algorithmic repercussions from operating with a large or, in the extreme case infinite, contact stiffness. Indeed, the algorithm used—a standard second-order cone programming solver—is independent of the particle scale model and is applicable to rigid as well as elastically deformable particles.

Large-Margin Estimation of Hidden Markov Models With Second-Order Cone Programming for Speech Recognition

Large-margin estimation (LME) holds a property of good generalization on unseen test data. In our previous work, LME of HMMs has been successfully applied to some small-scale speech recognition tasks, using the SDP (semi-definite programming) technique. In this paper, we further extend the previous work by exploring a more efficient convex optimization method with the technique of second-order cone programming (SOCP). More specifically, we have studied and proposed several SOCP relaxation techniques to convert LME of HMMs in speech recognition into a standard SOCP problem so that LME can be solved with more efficient SOCP methods. The formulation is general enough to deal with various types of competing hypothesis space, such as N-best lists and word graphs. The proposed LME/SOCP approaches have been evaluated on two standard speech recognition tasks. The experimental results on the TIDIGITS task show that the SOCP method significantly outperforms the gradient descent method, and achieve comparable performance with SDP, but with 20-200 times faster speed, requiring less memory and computing resources. Furthermore, the proposed LME/SOCP method has also been successfully applied to a large vocabulary task using the Wall Street Journals (WSJ0) database. The WSJ-5k recognition results show that the proposed method yields better performance than the conventional approaches including maximum-likelihood estimation (MLE), maximum mutual information estimation (MMIE), and more recent boosted MMIE methods.

Design of High-Order Extrapolated Impulse Response FIR Filters with Signed Powers-of-Two Coefficients

Expensive multiplication operations can be replaced by simpler additions and hardwired shifters so as to reduce power consumption and area size, if the coefficients of a digital filter are signed power-of-two (SPT). As a consequence, FIR digital filters with SPT coefficients have been widely studied in the last three decades. However, most approaches for the design of FIR filters with SPT coefficients focus on filters with length less than 100. These approaches are not suitable for the design of high-order filters because they require excessive computation time. In this paper, an approach for the design of high-order filters with SPT coefficients is proposed. It is a two-step approach. Firstly, the design of an extrapolated impulse response (EIR) filter is formulated as a standard second-order cone programming (SOCP) problem with an additional coefficient sensitivity constraint for optimizing its finite word-length effect. Secondly, the obtained continuous coefficients are quantized into SPT coefficients by recasting the filter-design problem into a weighted least squares (WLS) sequential quadratic programming relaxation (SQPR) problem. To further reduce implementation complexity, a graph-based common subexpression elimination (CSE) algorithm is utilized to extract common subexpressions between SPT coefficients. Simulation results show that the proposed method can effectively and efficiently design high-order SPT filters, including Hilbert transformers and half-band filters with SPT coefficients. Experiment results indicate that 0.81N∼0.29N adders are required for 18-bit N-order FIR filters (N=335∼3261) to meet the given magnitude response specifications.

Upper and lower bound limit analysis of plates using FEM and second-order cone programming

This paper presents two novel numerical procedures to determine upper and lower bounds on the actual collapse load multiplier for plates in bending. The conforming Hsieh-Clough-Tocher (HCT) and enhanced Morley (EM) elements are used to discrete the problem fields. A Morley element with enhanced moment fields is used. The constant moment fields is added a quadratic mode in which the pressure is equilibrated by corner loads only, ensuring that exact equilibrium relations associated with a uniform pressure can be obtained. Once the displacement or moment fields are approximated and the bound theorems applied, limit analysis becomes a problem of optimization. In this paper, the optimization problems are formulated in the form of a standard second-order cone programming which can be solved using highly efficient interior point solvers. The procedures are tested by applying it to several benchmark plate problems and are found good agreement between the present upper and lower bound solutions and results in the literature.

Upper bound limit analysis using simplex strain elements and second‐order cone programming

AbstractIn geomechanics, limit analysis provides a useful method for assessing the capacity of structures such as footings and retaining walls, and the stability of slopes and excavations. This paper presents a finite element implementation of the kinematic (or upper bound) theorem that is novel in two main respects. First, it is shown that conventional linear strain elements (6‐node triangle, 10‐node tetrahedron) are suitable for obtaining strict upper bounds even in the case of cohesive‐frictional materials, provided that the element sides are straight (or the faces planar) such that the strain field varies as a simplex. This is important because until now, the only way to obtain rigorous upper bounds has been to use constant strain elements combined with a discontinuous displacement field. It is well known (and confirmed here) that the accuracy of the latter approach is highly dependent on the alignment of the discontinuities, such that it can perform poorly if an unstructured mesh is employed. Second, the optimization of the displacement field is formulated as a standard second‐order cone programming (SOCP) problem. Using a state‐of‐the‐art SOCP code developed by researchers in mathematical programming, very large example problems are solved with outstanding speed. The examples concern plane strain and the Mohr–Coulomb criterion, but the same approach can be used in 3D with the Drucker–Prager criterion, and can readily be extended to other yield criteria having a similar conic quadratic form. Copyright © 2006 John Wiley & Sons, Ltd.