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.

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.

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.

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.

Nonsingularity in second-order cone programming via the smoothing metric projector

Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point.Moreover,this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point.Interestingly,the analysis does not need the strict complementarity condition.

Exact Computable Representation of Some Second-Order Cone Constrained Quadratic Programming Problems

Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation,which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints.In this paper,we provide exact computable representations for some more subclasses of the QCQP problem,in particular,the subclass with one secondorder cone constraint and two special linear constraints.

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.

Unified convergence analysis of a second-order method of multipliers for nonlinear conic programming

In this paper,we accomplish the unified convergence analysis of a second-order method of multipliers(i.e.,a second-order augmented Lagrangian method)for solving the conventional nonlinear conic optimization problems.Specifically,the algorithm that we investigate incorporates a specially designed nonsmooth(generalized)Newton step to furnish a second-order update rule for the multipliers.We first show in a unified fashion that under a few abstract assumptions,the proposed method is locally convergent and possesses a(nonasymptotic)superlinear convergence rate,even though the penalty parameter is fixed and/or the strict complementarity fails.Subsequently,we demonstrate that for the three typical scenarios,i.e.,the classic nonlinear programming,the nonlinear second-order cone programming and the nonlinear semidefinite programming,these abstract assumptions are nothing but exactly the implications of the iconic sufficient conditions that are assumed for establishing the Q-linear convergence rates of the method of multipliers without assuming the strict complementarity.

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.

Quadratic Optimization over a Second-Order Cone with Linear Equality Constraints

This paper studies the nonhomogeneous quadratic programming problem over a second-order cone with linear equality constraints.When the feasible region is bounded,we show that an optimal solution of the problem can be found in polynomial time.When the feasible region is unbounded,a semidefinite programming(SDP)reformulation is constructed to find the optimal objective value of the original problem in polynomial time.In addition,we provide two sufficient conditions,under which,if the optimal objective value is finite,we show the optimal solution of SDP reformulation can be decomposed into the original space to generate an optimal solution of the original problem in polynomial time.Otherwise,a recession direction can be identified in polynomial time.Numerical examples are included to illustrate the effectiveness of the proposed approach.

Multiple-constraint cooperative guidance based on two-stage sequential convex programming

An improved approach is presented in this paper to implement highly constrained cooperative guidance to attack a stationary target.The problem with time-varying Proportional Navigation(PN)gain is first formulated as a nonlinear optimal control problem,which is difficult to solve due to the existence of nonlinear kinematics and nonconvex constraints.After convexification treatments and discretization,the solution to the original problem can be approximately obtained by solving a sequence of Second-Order Cone Programming(SOCP)problems,which can be readily solved by state-of-the-art Interior-Point Methods(IPMs).To mitigate the sensibility of the algorithm on the user-provided initial profile,a Two-Stage Sequential Convex Programming(TSSCP)method is presented in detail.Furthermore,numerical simulations under different mission scenarios are conducted to show the superiority of the proposed method in solving the cooperative guidance problem.The research indicated that the TSSCP method is more tractable and reliable than the traditional methods and has great potential for real-time processing and on-board implementation.

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.

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.

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.

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.

Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity

The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are established. Finally, a self duality theorem is given.

Convexification of Hybrid AC-DC Optimal Power Flow with Line-Commutated Converters

Line-commutated converter (LCC)-based high-voltage DC (HVDC) systems have been integrated with bulk AC power grids for interregional transmission of renewable power. The nonlinear LCC model brings additional nonconvexity to optimal power flow (OPF) of hybrid AC-DC power grids. A convexification method for the LCC station model could address such nonconvexity but has rarely been discussed. We devise an equivalent reformulation for classical LCC station models that facilitates second-order cone convex relaxation for the OPF of LCC-based AC-DC power grids. We also propose sufficient conditions for exactness of convex relaxation with its proof. Equivalence of the proposed LCC station models and properties, exactness, and effectiveness of convex relaxation are verified using four numerical simulations. Simulation results demonstrate a globally optimal solution of the original OPF can be efficiently obtained from relaxed model.

Exploiting demand-side heterogeneous flexible resources in risk management of power system frequency

More demand-side flexible resources(DFRs)are participating in the frequency regulation of renewable power systems,whose heterogeneous characteristics have a significant impact on the system frequency response.Consequently,selecting suitable DFRs poses a formidable challenge for independent system operators(ISO).In this paper,a reserve allocation methodology for heterogeneous DFRs is proposed to manage the risk of power system frequency.Firstly,a performance curve is developed to describe the cost,capacity,and response speed of DFRs.Moreover,a clustering method for multiple distributed DFRs is conducted to calculate the aggregated performance curves and uncertainty coefficients.Then,the frequency security criterion considering DFRs’performance is constructed,whose linearity makes it can be easily coupled into the system scheduling model and solved.Furthermore,a risk management model for DFRs considering frequency-chance-constraint is proposed to make a trade-off between cost and frequency security.Finally,the model is transformed into mixed integer second-order cone programming(MISOCP)and solved by the commercial solver.The proposed model is validated by the IEEE 30 and IEEE 118 bus systems.