Anti-interference beam pattern design based on second-order cone programming optimization

When signal-to-interference ratio is low, the energy of strong interference leaked from the side lobe of beam pattern will infect the detection of weak target. Therefore, the beam pattern needs to be optimized. The existing Dolph-Chebyshev weighting method can get the lowest side lobe level under given main lobe width, but for the other non-uniform circular array and nonlinear array, the low side lobe pattern needs to be designed specially. The second order cone programming optimization （SOCP） algorithm proposed in the paper transforms the optimization of the beam pattern into a standard convex optimization problem. Thus there is a paradigm to follow for any array formation, which not only achieves the purpose of Dolph-Chebyshev weighting, but also solves the problem of the increased side lobe when the signal is at end fire direction The simulation proves that the SOCP algorithm can detect the weak target better than the conventional beam forming.

Improved design of reconfigurable frequency response masking filters based on second-order cone programming

In order to improve the design results for the reconfigurable frequency response masking FRM filters an improved design method based on second-order cone programming SOCP is proposed.Unlike traditional methods that separately design the proposed method takes all the desired designing modes into consideration when designing all the subfilters. First an initial solution is obtained by separately designing the subfilters and then the initial solution is updated by iteratively solving a SOCP problem. The proposed method is evaluated on a design example and simulation results demonstrate that jointly designing all the subfilters can obtain significantly lower minimax approximation errors compared to the conventional design method.

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.

Robust Blind Separation for MIMO Systems against Channel Mismatch Using Second-Order Cone Programming

To improve the deteriorated capacity gain and source recovery performance due to channel mismatch problem,this paper reports a research about blind separation method against channel mismatch in multiple-input multiple-output(MIMO) systems.The channel mismatch problem can be described as a channel with bounded fluctuant errors due to channel distortion or channel estimation errors.The problem of blind signal separation/extraction with channel mismatch is formulated as a cost function of blind source separation(BSS) subject to the second-order cone constraint,which can be called as second-order cone programing optimization problem.Then the resulting cost function is solved by approximate negentropy maximization using quasi-Newton iterative methods for blind separation/extraction source signals.Theoretical analysis demonstrates that the proposed algorithm has low computational complexity and improved performance advantages.Simulation results verify that the capacity gain and bit error rate(BER) performance of the proposed blind separation method is superior to those of the existing methods in MIMO systems with channel mismatch problem.

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.

On Second-Order Duality in Nondifferentiable Continuous Programming

A Mond-Weir type second-order dual continuous programming problem associated with a class of nondifferentiable continuous programming problems is formulated. Under second-order pseudo-invexity and second-order quasi-invexity various duality theorems are established for this pair of dual continuous programming problems. A pair of dual continuous programming problems with natural boundary values is constructed and the proofs of its various duality results are briefly outlined. Further, it is shown that our results can be regarded as dynamic generalizations of corresponding (static) second-order duality theorems for a class of nondifferentiable nonlinear programming problems already studied in the literature.

Second-Order Duality for Continuous Programming Containing Support Functions

A second-order dual problem is formulated for a class of continuous programming problem in which both objective and constrained functions contain support functions, hence it is nondifferentiable. Under second-order invexity and second-order pseudoinvexity, weak, strong and converse duality theorems are established for this pair of dual problems. Special cases are deduced and a pair of dual continuous problems with natural boundary values is constructed. A close relationship between duality results of our problems and those of the corresponding (static) nonlinear programming problem with support functions is briefly outlined.

Optimality Conditions and Second-Order Duality for Nondifferentiable Multiobjective Continuous Programming Problems

Fritz John and Karush-Kuhn-Tucker type optimality conditions for a nondifferentiable multiobjective variational problem are derived. As an application of Karush-Kuhn-Tucker type optimality conditions, Mond-weir type second-order nondifferentiable multiobjective dual variational problems is constructed. Various duality results for the pair of Mond-Weir type second-order dual variational problems are proved under second-order pseudoinvexity and second-order quasi-invexity. A pair of Mond-Weir type dual variational problems with natural boundary values is formulated to derive various duality results. Finally, it is pointed out that our results can be considered as dynamic generalizations of their static counterparts existing in the literature.

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.

On Continuous Programming with Support Functions

A second-order Mond-Weir type dual problem is formulated for a class of continuous programming problems in which both objective and constraint functions contain support functions;hence it is nondifferentiable. Under second-order strict pseudoinvexity, second-order pseudoinvexity and second-order quasi-invexity assumptions on functionals, weak, strong, strict converse and converse duality theorems are established for this pair of dual continuous programming problems. Special cases are deduced and a pair of dual continuous problems with natural boundary values is constructed. A close relationship between the duality results of our problems and those of the corresponding (static) nonlinear programming problem with support functions is briefly outlined.

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.

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.

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.

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.

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.

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.