SMOOTHING NEWTON ALGORITHM FOR THE CIRCULAR CONE PROGRAMMING WITH A NONMONOTONE LINE SEARCH

In this paper, we present a nonmonotone smoothing Newton algorithm for solving the circular cone programming(CCP) problem in which a linear function is minimized or maximized over the intersection of an affine space with the circular cone. Based on the relationship between the circular cone and the second-order cone(SOC), we reformulate the CCP problem as the second-order cone problem(SOCP). By extending the nonmonotone line search for unconstrained optimization to the CCP, a nonmonotone smoothing Newton method is proposed for solving the CCP. Under suitable assumptions, the proposed algorithm is shown to be globally and locally quadratically convergent. Some preliminary numerical results indicate the effectiveness of the proposed algorithm for solving the CCP.

Adaptive broadband beamformer for nonuniform linear array based on second order cone programming

Adaptive broadband beamforraing is a key issue in array applications. The adaptive broadband beamformer with tapped delay line （TDL） structure for nonuniform linear array （NLA） is designed according to the rule of minimizing the beamformer＇s output power while keeping the distortionless response （DR） in the direction of desired signal and keeping the constant beamwidth （CB） with the prescribed sidelobe level over the whole operating band. This kind of beamforming problem can be solved with the interior-point method after being converted to the form of standard second order cone programming （SOCP）. The computer simulations are presented which illustrate the effectiveness of our beamformer.

Two new predictor-corrector algorithms for second-order cone programming

Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming （SOCP） are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton （AHO） directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O（rln（εo/ε）） is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.

A VU-decomposition method for a second-order cone programming problem

A vu-decomposition method for solving a second-order cone problem is presented in this paper. It is first transformed into a nonlinear programming problem. Then, the structure of the Clarke subdifferential corresponding to the penalty function and some results of itsvu-decomposition are given. Under a certain condition, a twice continuously differentiable trajectory is computed to produce a second-order expansion of the objective function. A conceptual algorithm for solving this problem with a superlinear convergence rate is given.

Approximation of the Shannon Capacity Via Matrix Cone Programming

This paper proposes a novel formulation using the matrix cone programming to compute an upper bound of the Shannon capacity of graphs,which is theoretically superior to the Lovász number.To achieve this,a sequence of matrix cones is constructed by adding certain co-positive matrices to the positive semi-definite matrix cones during the matrix cone programming.We require the sequence of matrix cones to have the weak product property so that the improved result of the matrix cone programming remains an upper bound of the Shannon capacity.Our result shows that the existence of a sequence of suitable matrix cones with the weak product property is equivalent to the existence of a co-positive matrix with testable conditions.Finally,we give some concrete examples with special structures to verify the existence of the matrix cone sequence.

Day-ahead Optimization Schedule for Gas-electric Integrated Energy System Based on Second-order Cone Programming

This paper proposes an optimal day-ahead opti-mization schedule for gas-electric integrated energy system(IES)considering the bi-directional energy flow.The hourly topology of electric power system(EPS),natural gas system(NGS),energy hubs(EH)integrated power to gas(P2G)unit,are modeled to minimize the day-ahead operation cost of IES.Then,a second-order cone programming(SOCP)method is utilized to solve the optimization problem,which is actually a mixed integer nonconvex and nonlinear programming issue.Besides,cutting planes are added to ensure the exactness of the global optimal solution.Finally,simulation results demonstrate that the proposed optimization schedule can provide a safe,effective and economical day-ahead scheduling scheme for gas-electric IES.

Optimal design and verification of temporal and spatial filters using second-order cone programming approach

Temporal filters and spatial filters are widely used in many areas of signal processing. A number of optimal design criteria to these problems are available in the literature. Various computational techniques are also presented to optimize these criteria chosen. There are many drawbacks in these methods. In this paper, we introduce a unified framework for optimal design of temporal and spatial filters. Most of the optimal design problems of FIR filters and beamformers are included in the framework. It is shown that all the design problems can be reformulated as convex optimization form as the second-order cone programming （SOCP） and solved efficiently via the well-established interior point methods. The main advantage of our SOCP approach as compared with earlier approaches is that it can include most of the existing methods as its special cases, which leads to more flexible designs. Furthermore, the SOCP approach can optimize multiple required performance measures, which is the drawback of earlier approaches. The SOCP approach is also developed to optimally design temporal and spatial two-dimensional filter and spatial matrix filter. Numerical results demonstrate the effectiveness of the proposed approach.

On Iteration Complexity of a First-Order Primal-Dual Method for Nonlinear Convex Cone Programming

Nonlinear convex cone programming(NCCP)models have found many practical applications.In this paper,we introduce a flexible first-order primal-dual algorithm,called the variant auxiliary problem principle(VAPP),for solving NCCP problems when the objective function and constraints are convex but may be nonsmooth.At each iteration,VAPP generates a nonlinear approximation of the primal augmented Lagrangian model.The approximation incorporates both linearization and a distance-like proximal term,and then the iterations of VAPP are shown to possess a decomposition property for NCCP.Motivated by recent applications in big data analytics,there has been a growing interest in the convergence rate analysis of algorithms with parallel computing capabilities for large scale optimization problems.We establish O(1/t)convergence rate towards primal optimality,feasibility and dual optimality.By adaptively setting parameters at different iterations,we show an O(1/t2)rate for the strongly convex case.Finally,we discuss some issues in the implementation of VAPP.

A New Infeasible-Interior-Point Algorithm Based on Wide Neighborhoods for Symmetric Cone Programming

In this paper,we present an infeasible-interior-point algorithm,based on a new wide neighborhood for symmetric cone programming.We treat the classical Newton direction as the sum of two other directions,and equip them with different step sizes.We prove the complexity bound of the new algorithm for the Nesterov-Todd(NT)direction,and the xs and sx directions.The complexity bounds obtained here are the same as small neighborhood infeasible-interior-point algorithms over symmetric cones.

Polynomial Convergence of Primal-Dual Path-Following Algorithms for Symmetric Cone Programming Based on Wide Neighborhoods and a New Class of Directions

This paper presents a class of primal-dual path-following interior-point algorithms for symmetric cone programming(SCP)based on wide neighborhoods and new directions with a parameterθ.When the parameterθ=1,the direction is exactly the classical Newton direction.When the parameterθis independent of the rank of the associated Euclidean Jordan algebra,the algorithm terminates in at most O(κr logε−1)iterations,which coincides with the best known iteration bound for the classical wide neighborhood algorithms.When the parameterθ=√n/βτand Nesterov–Todd search direction is used,the algorithm has O(√r logε−1)iteration complexity,the best iteration complexity obtained so far by any interior-point method for solving SCP.To our knowledge,this is the first time that a class of interior-point algorithms including the classical wide neighborhood path-following algorithm is proposed and analyzed over symmetric cone.

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.

Effect of eccentric and inclined loading on the bearing capacity of strip footing placed on rock mass

This paper deals with the bearing capacity determination of strip footing on a rock mass in hilly area by considering the influence of inclined and eccentric loading. Applying the generalized HoekBrown failure criterion, the failure behavior of the rock mass is modeled with the help of the power cone programming in the lower bound finite element limit analysis framework. Using bearing capacity factor(Ns), the change in bearing capacity of the strip footing due to the occurrence of eccentrically inclined loading is presented. The variations of the magnitude of Ns are obtained by examining the effects of the Hoek-Brown rock mass strength parameters(uniaxial compressive strength(sci), disturbance factor(D), rock parameter(mi), and Geological Strength Index(GSI)) in the presence of different magnitudes of eccentricity(e) and inclination angle(λ) with respect to the vertical plane, and presented as design charts. Both the inclined loading modes, i.e., inclination towards the center of strip footing(+λ) and inclination away from the center of strip footing(-λ), are adopted to perform the investigation. In addition, the correlation between the input parameters and the corresponding output is developed by utilizing the artificial neural network(ANN). Additionally, from sensitivity analysis, it is observed that inclination angle(λ) is the most sensitive parameter. For practicing engineers, the obtained design equation and design charts can be beneficial to understand the bearing capacity variation in the existence of eccentrically inclined loading in mountain areas.

AN INFEASIBLE-INTERIOR-POINT PREDICTOR-CORRECTOR ALGORITHM FOR THE SECOND-ORDER CONE PROGRAM

A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.

Strain localization of Mohr-Coulomb soils with non-associated plasticity based on micropolar continuum theory

To address the problems of strain localization, the exact Mohr-Coulomb (MC) model is used based on second-order cone programming (mpcFEM-SOCP) in the framework of micropolar continuum finite element method. Using the uniaxial compression test, we focused on the earth pressure problem of rigid wall segment involving non-associated plasticity. The numerical results reveal that when mpcFEM-SOCP is applied, the problems of mesh dependency can be effectively addressed. For geotechnical strain localization analysis involving non-associated MC plasticity, mpcFEM-SOCP in conjunction with the pseudo-time discrete scheme can improve the numerical stability and avoid the unreasonable softening issue in the pressure-displacement curves, which may be encountered in the conventional FEM. It also shows that the pressure-displacement responses calculated by mpcFEM-SOCP with the pseudo-time discrete scheme are higher than those calculated by mpcFEM-SOCP with the Davis scheme. The inclination angle of shear band predicted by mpcFEM-SOCP with the pseudo-time discrete scheme agrees well with the theoretical solution of non-associated MC plasticity.

Multiconstraint adaptive three-dimensional guidance law using convex optimization

The traditional guidance law only guarantees the accuracy of attacking a target. However, the look angle and acceleration constraints are indispensable in applications. A new adaptive three-dimensional proportional navigation(PN) guidance law is proposed based on convex optimization. Decomposition of the three-dimensional space is carried out to establish threedimensional kinematic engagements. The constraints and the performance index are disposed by using the convex optimization method. PN guidance gains can be obtained by solving the optimization problem. This solution is more rapid and programmatic than the traditional method and provides a foundation for future online guidance methods, which is of great value for engineering applications.

The use of the node-based smoothed finite element method to estimate static and seismic bearing capacities of shallow strip footings

The node-based smoothed finite element method(NS-FEM)is shortly presented for calculations of the static and seismic bearing capacities of shallow strip footings.A series of computations has been performed to assess variations in seismic bearing capacity factors with both horizontal and vertical seismic accelerations.Numerical results obtained agree very well with those using the slip-line method,revealing that the magnitude of the seismic bearing capacity is highly dependent upon the combinations of various directions of both components of the seismic acceleration.An upward vertical seismic acceleration reduces the seismic bearing capacity compared to the downward vertical seismic acceleration in calculations.In addition,particular emphasis is placed on a separate estimation of the effects of soil and superstructure inertia on each seismic bearing capacity component.While the effect of inertia forces arising in the soil on the seismic bearing capacity is non-trivial,and the superstructure inertia is the major contributor to reductions in the seismic bearing capacity.Both tables and charts are given for practical application to the seismic design of the foundations.

SRV constraint based FIB design for wideband linear array

Frequency-invariant beamformer （FIB） design is a key issue in wideband array signal processing. To use commonly wideband linear array with tapped delay line （TDL） structure and complex weights, the FIB design is provided according to the rule of minimizing the sidelobe level of the beampattern at the reference frequency while keeping the distortionless response constraint in the mainlobe direction at the reference frequency, the norm constraint of the weight vector and the amplitude constraint of the averaged spatial response variation （SRV）. This kind of beamformer design problem can be solved with the interior-point method after being converted to the form of standard second order cone programming （SOCP）. The computer simulations are presented which illustrate the effectiveness of our FIB design method for the wideband linear array with TDL structure and complex weights.

Directional Source Localization Based on RSS-AOA Combined Measurements

Source localization plays an indispensable role in many applications.This paper addresses the directional source localization problem in a three-dimensional(3D)wireless sensor network using hybrid received-signal-strength(RSS)and angle-of-arrival(AOA)measurements.Both the position and transmission orientation of the source are to be estimated.In the considered positioning scenario,the angle and range measurements are respectively corresponding to the AOA model and RSS model that integrates the Gaussian-shaped radiation pattern.Given that the localization problem is non-convex and the unknown parameters therein are coupled together,this paper adopts the second-order cone relaxation and alternating optimization techniques in the proposed estimation algorithm.Moreover,to provide a performance benchmark for any localization method,the corresponding Cramer-Rao lower bounds(CRLB)of estimating the unknown position and transmission orientation of the source are derived.Numerical and simulation results demonstrate that the presented algorithm effectively resolves the problem,and its estimation performance is close to the CRLB for the localization with the hybrid measurements.

A stable CS-FEM for the static and seismic stability of a single square tunnel in the soil where the shear strength increases linearly with depth

A numerical procedure using a stable cell-based smoothed finite element method(CS-FEM)is presented for estimation of stability of a square tunnel in the soil where the shear strength increases linearly with depth.The kinematically admissible displacement fields are approximated by uniform quadrilateral elements in conjunction with the strain smoothing technique,eliminating volumetric locking issues and the singularity associated with the MohreCoulomb model.First,a rich set of simulations was performed to compute the static stability of a square tunnel with different geometries and soil conditions.The presented results are in excellent agreement with the upper and lower bound solutions using the standard finite element method(FEM).The stability charts and tables are given for practical use in the tunnel design,along with a newly proposed formulation for predicting the undrained stability of a single square tunnel.Second,the seismic stability number was computed using the present numerical approach.Numerical results reveal that the seismic stability number reduces with an increasing value of the horizontal seismic acceleration(a_(h)),for both cases of the weightless soil and the soil with unit weight.Third,the link between the static and seismic stability numbers is described using corrective factors that represent reductions in the tunnel stability due to seismic loadings.It is shown from the numerical results that the corrective factor becomes larger as the unit weight of soil mass increases;however,the degree of the reduction in seismic stability number tends to reduce for the case of the homogeneous soil.Furthermore,this advanced numerical procedure is straightforward to extend to three-dimensional(3D)limit analysis and is readily applicable for the calculation of the stability of tunnels in highly anisotropic and heterogeneous soils which are often encountered in practice.

Seismic bearing capacity of a strip footing on rock media

The bearing capacity factors for a rough strip footing placed on rock media,which is subjected to pseudostatic horizontal earthquake body forces,have been determined using the lower bound finite element limit analysis in conjunction with the power cone programming(PCP).The rock mass is assumed to follow the generalized Hoek-Brown(GHB)yield criterion.No assumption needs to be made to smoothen the GHB yield criterion and the convergence is found to achieve quite rapidly while performing the optimization with the usage of the PCP.While incorporating the variation in horizontal earthquake acceleration coefficient(kh),the effect of changes in unit weight of rock mass(γ),ground surcharge pressure(q0)and the associated GHB material shear strength parameters(geological strength index(GSI),yield parameter(mi),uniaxial compressive strength(σci))on the bearing capacity factors has been thoroughly assessed.Non-dimensional charts have been developed for design purpose.The accuracy of the present analysis has been duly checked by comparing the obtained results with the different solutions reported in the literature.The failure patterns have also been examined in detail.