Research On The Design Of Binary Sequence Sets Based On Consensus-ADMM Algorithm

Alternating Direction Method of Multipliers(ADMM)is an effective method for solving distributed convex optimization problems.It combines the augmented Lagrangian function and the dual ascent method.It can be widely applied in machine and statistical learning,compression sensing,signal processing of mass data processing field.There are some general achievements in the convergence performance of ADMM algorithm and its improved form for solving convex optimization problems.But the related theoretical results for solving non-convex optimization problems are not general.However,the optimization problems in practical applications,such as machine and statistical learning,signal and image processing,compression sensing and other field are mostly non-convex.How to design ADMM solution algorithm with theoretically-guaranteed performance and high execution efficiency combining with the internal structure of practical application problems has very clear practical significance.The constant modulus sequence with good correlation characteristics is widely used in wireless communication and MIMO radar systems.It can distinguish users and channels,reduce interference between users,improve channel capacity,control the frequency/space energy allocation of radar systems,and improve radar systems the ability of Antiinterception.Designing binary sequences with the above purposes attracts the attention of many scholars.Since the phases of the sequence are distributed on discrete points in the unit circle,the problem of binary sequence design is a classic NP-hard problem.In order to solve the problem,most of the existing heuristic methods convert the original design problem into a low-order problem,and then solve it according to certain rules and hard decisions.These processing methods lack theoretical support and cannot guarantee high-quality solutions.In addition,the design problem is often a high-order polynomial of sequence variables.The existing methods use the serial solution framework with low execution efficiency,which makes it difficult to deal with large-scale sequence set design problems.In this thesis,the penalty term method and lp-box method are used to deal with the discrete constraints of the low correlation binary sequence design problem.A low complexity and theoretically guaranteed consensus ADMM algorithm is proposed to solve the problem,and the convergence and computational complexity of the algorithm are studied.The main contents and work of this paper are as follows:Firstly,in the thesis,according to the demand of wireless communication and radar system for low-correlation characteristics binary sequence,the optimization model is established.Since the objective function of the formulated model is a quartic non-convex polynomial about the sequence set,it is difficult to solve directly;the binary constraints are discrete,so the whole model becomes a combination optimization problem of variables,and the computational complexity of the problem will increase exponentially with the increase of sequence set,which is difficult to accept in practice.Therefore,how to deal with discrete poly-phase constraints and ensure the convergence of the algorithm are key scientific problems of sequence design.In this thesis,penalty term method and lp-box method are used to deal with binary constraints respectively.Penalty term method refers to relaxing binary discrete constraints in the model to box constraints.In order to avoid the effect of this operation,such zero value will appear in the update process,so that the effect of correct solution is not obtained at last,consider adding a quadratic penalty term to the objective function to tighten the relaxation.The lp-box method refers to equivalently replacing discrete binary constraints with a series of continuous constraints,that is,relaxing the binary constraints to lp-box.The above treatment of discrete constraints can not only avoid the pain points which are difficult to solve,but also can use the mature technology in the field of continuous optimization for solving approximate problems.Compared with the existing methods,it should have better performance guarantee.Secondly,based on the internal structure of the optimization problem,auxiliary variables are introduced to make it equivalent to a consensus optimization problem.This thesis proposes consensus ADMM algorithm to approximately solve this consensus problem.It should be emphasized that the reformulated problem split into multiple sub-problems about local variables and can be solved distributed among each sub-problem.Therefore,the designed ADMM algorithm can be suitable for the design optimization of large scale binary sequences.When solving each sub-problem in detail,fully exploring and utilizing the sparse structure of the problem can reduce the computational complexity and greatly improve the efficiency of the algorithm.Finally,the performance of the proposed distributed algorithm including convergence,local optimality and computational complexity is analyzed theoretically.The simulation results show that the proposed algorithm,compared with the existing binary sequence design algorithm,has better performance and can set the correlation interval flexibly.