Performance analysis of LVI-based PDNN applied to real-time solution of time-varying quadratic programming

Abstract

This paper illustrates theoretical analysis and simulative verification on the performance of the linear-variationalinequality based primal-dual neural network (LVI-PDNN), which was designed originally for static quadratic programming (QP) problem solving but is now applied to time-varying QP problem solving. It is theoretically proved that the LVI-PDNN for solving the time-varying QP problem subject to equality, inequality and bound constraints simultaneously could only approximately approach the time-varying theoretical solution, instead of converging exactly. In other words, the steady-state error of the realtime solution can not decrease to zero. In order to better evaluate the time-varying situation, we investigate the upper bound of such an error and the global exponential convergence rate for the LVI-PDNN approaching its loose error bound. Computer simulations further substantiate the performance analysis of the LVI-PDNN exploited for real-time solution of the time-varying QP problem. I. TIME-VARYING QP OF INTERESTQuadratic programming (QP) problems play a significant role in mathematical optimization, and have been theoretically analyzed [1][2] and extensively applied to plenty of scientific fields; e.g., optimal controller design, power-scheduling, digital signal processing, and robot-arm motion planning [3][4]. In the past, researchers usually handle optimization problems only subject to one or two kinds of constraints [5]. In addition, some QP problems are just investigated based on static coefficients (or to say, constant coefficient matrices and vectors) [6], which may not applicative for time-varying cases. Motivated by realtime engineering applications in robotics [5][7], the general time-varying QP (TVQP) in this paper is presented as follows.where Hessian matrix W (t) ∈ R n×n is smoothly timevarying, positive-definite and symmetric at any time instant t ∈ [0, +∞). Besides, coefficient matrices J(t) ∈ R m×n and A(t) ∈ R k×n as well as coefficient vectors q(t) ∈ R n , ξ − (t) ∈ R n , ξ + (t) ∈ R n , d(t) ∈ R m and b(t) ∈ R k are all assumed smoothly time-varying. In time-varying QP (1)-(4), unknown vector x(t) ∈ R n is to be solved in real time t ∈ [0, +∞). II. GENERAL SOLUTIONS TO STATIC QPTo solve the fundamental static QP problem, a lot of methods/algorithms have been proposed [1] [2]. In general, there are two common solutions to such a QP problem. The first one is the numerical algorithms performed on digital computers and it has been widely used to solve small-scale static QP problems. However, when it comes to large-scale real-time applications, in view of its serial-processing nature, such numerical algorithms may result in decline of the performances [8]. Usually, the minimal arithmetic operations are proportional to the cube of Hessian matrix dimension n, which is computationally expensive. As for the second general type of solution, the application of parallel processing has influenced the algorithmic developments [9][10]. Thus, various dynamic and analog solvers have been developed and ...

Time-varying mean–variance portfolio selection problem solving via LVI-PDNN

“…This problem can also be written in the time-varying 186 quadratic programming (TVQP) problem form, by following 187 [21], as follows:…”

“…. , 0] ∈ 200 R n denotes the zero vector and (5)- (7) to the coefficients of the LVI-PDNN from [21]. Accord-…”

“…Furthermore, the definition of the primal-dual decision vector 235 y(t) can be written as follows, along with the lower and upper 236 boundaries to which it is subject: 12)-( 15) to the coefficients of the LVI-PDNN from [21].…”

“…where P Ω (•) is the projection operator (see [21]) and γ > 0 is known as the design parameter. Within hardware permission,…”

“…In particular, May, July, August have 22 observations each, June has 20 observations, September and October have 21 observations together. So, we have tspan = [0 5] and by setting noep = [22,20,22,22,21] as input in the function omega, we get…”