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(ε0/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.

Feasibility and Structural Feature on Monotone Second-Order Cone Linear Complementarity Problems in Hilbert Space

Given a real finite-dimensional or infinite-dimensional Hilbert space H with a Jordan product, the second-order cone linear complementarity problem(SOCLCP)is considered. Some conditions are investigated, for which the SOCLCP is feasible and solvable for any element q?H. The solution set of a monotone SOCLCP is also characterized. It is shown that the second-order cone and Jordan product are interconnected.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

New existence results are presented for the singular second_order nonlinear boundary value problems u″+g(t)f(u)=0, 0<t<1, αu(0)-βu′(0)=0, γu(1)+ δu′(1)=0under the conditions0≤f + 0<M 1, m 1<f - ∞≤∞ or0≤f+ ∞<M 1, m 1<f- 0≤∞, where f+ 0= lim u→0 f(u)/u, f - ∞= lim u→∞ f(u)/u, f - 0= lim u→0 f(u)/u, f + ∞= lim u→∞ f(u)/u, g may be singular at t=0 and/or t=1 The proof uses a fixed point theorem in cone theory.

Dispatchable Region for Active Distribution Networks Using Approximate Second-order Cone Relaxation

Uncertainty in distributed renewable generation threatens the security of power distribution systems.The concept of dispatchable region is developed to assess the ability of power systems to accommodate renewable generation at a given operating point.Although DC and linearized AC power flow equations are typically used to model dispatchable regions for transmission systems,these equations are rarely suitable for distribution networks.To achieve a suitable trade-off between accuracy and efficiency,this paper proposes a dispatchable region formulation for distribution networks using tight convex relaxation.Secondorder cone relaxation is adopted to reformulate AC power flow equations,which are then approximated by a polyhedron to improve tractability.Further,an efficient adaptive constraint generation algorithm is employed to construct the proposed dispatchable region.Case studies on distribution systems of various scales validate the computational efficiency and accuracy of the proposed method.

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.

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.

Second-Order Optimality Conditions for Multiobjective Optimization Whose Order Induced by Second-Order Cone

This paper is devoted to developing first-order necessary,second-order necessary,and second-order sufficient optimality conditions for a multiobjective optimization problem whose order is induced by a finite product of second-order cones(here named as Q-multiobjective optimization problem).For an abstract-constrained Q-multiobjective optimization problem,we derive two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions.For Q-multiobjective optimization problem with explicit constraints,we demonstrate first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as second-order sufficient optimality conditions under upper second-order regularity for the explicit constraints.As applications,we obtain optimality conditions for polyhedral conic,second-order conic,and semi-definite conic Q-multiobjective optimization problems.

收益率椭球分布不确定下的均值-CVaR优化研究

The article explores a mean-CVaR ratio model with returns distribution uncertainty.To describe the uncertainty of returns distribution,a mixture ellipsoidal distribution absorbing some typical distributions such as the mixture distribution and and ellipsoidal distribution is introduced.Then,by using robust technique with some assumptions,the original robust mean-CVaR ratio model can be formulated as a second-order cone optimization model where the underlying random returns have a mixture ellipsoidal distribution.As an illustration,the corresponding robust optimization models are applied to allocations of assets in securities market.Numerical simulations are presented to illustrate the relation between robustness and optimality and to compare mixture ellipsoidal distribution to some typical distributions as well.

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.

Convergence of an augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Lowner operator associated with a potential function for the optimization problems with inequality constraints.The favorable properties of both the Lowner operator and the corresponding augmented Lagrangian are discussed.And under some mild assumptions,the rate of convergence of the augmented Lagrange algorithm is studied in detail.

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.

A Modified and Simplified Full Nesterov–Todd Step O(N)Infeasible Interior-Point Method for Second-Order Cone Optimization

We present a modified and simplified version of an infeasible interior-point method for second-order cone optimization published in 2013(Zangiabadi et al.in J Optim Theory Appl,2013).In the earlier version,each iteration consisted of one socalled feasibility step and a few centering steps.Here,each iteration consists of only a feasibility step.Thus,the new algorithm improves the number of iterations and the improvement is due to a lemma which gives an upper bound for the proximity after the feasibility step.The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

A Class of Second-Order Cone Eigenvalue Complementarity Problems for Higher-Order Tensors

In this paper,we consider the second-order cone tensor eigenvalue complementarity problem(SOCTEiCP)and present three different reformulations to the model under consideration.Specifically,for the general SOCTEiCP,we first show its equivalence to a particular variational inequality under reasonable conditions.A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem.Then,for the symmetric and sub-symmetric SOCTEiCPs,we reformulate them as appropriate nonlinear programming problems,which are extremely beneficial for designing reliable solvers to find solutions of the considered problem.Finally,we report some preliminary numerical results to verify our theoretical results.

A New Complementarity Function and Applications in Stochastic Second-Order Cone Complementarity Problems

This paper considers the so-called expected residual minimization(ERM)formulation for stochastic second-order cone complementarity problems,which is based on a new complementarity function called termwise residual complementarity function associated with second-order cone.We show that the ERM model has bounded level sets under the stochastic weak R0-property.We further derive some error bound results under either the strong monotonicity or some kind of constraint qualifications.Then,we apply the Monte Carlo approximation techniques to solve the ERM model and establish a comprehensive convergence analysis.Furthermore,we report some numerical results on a stochastic second-order cone model for optimal power flow in radial networks.

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.

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.