subproblems involving the solution of linear equations, which lend themselves to the use of memristors for efficient computation. A. Linear Optimization With Memristors The standard form of LP is expressed as follows, minimize dT x x subject to Gx = h, (10) n where x ! R is the optimization variable, d ! R , G ! R l # n and h ! R l are given parameters, and the last inequality constraint represents the elementwise inequalities x i $ 0 for i = 1, 2, f, n. In this paper, we assume that G is of full row rank. We begin by reformulating problem (10) as the canonical form (4) that is amenable to the use of ADMM algorithm, d T x + p (x) + g (y) minimize x,y (11) subject to x = y, where y ! R n is a newly introduced optimization variable, and similar to (5), p and g are indicator functions, with respect to constraint sets {x|Gx = h} and {y | y $ 0}, respectively. If we set f (x) = d T x + p (x), A = I, B =-I and c = 0 , then problem (11) is the same as problem (4). Based on (11), the ADMM steps (6)-(9) become x k +1 = arg min $ d T x + p (x) + (n k) T (x - y k) x + t 2 2 x - yk 2 . (12) y k +1 = arg min $ g (y) + (n k ) T (x k +1 - y) y + t 2 2 x k +1 - y 2 . (13) n k +1 = n k + t (x k +1 - y k +1) . (14) As we show next, the primary advantage of employing ADMM here is that problem (12) can be readily solved using memristor crossbars, and problem (13) yields a closed-form solution that only involves elementary vector operations. Problem (12) is equivalent to t ||x - a||22 2 subject to Gx = h, minimize x (15) where a := y k - (1/t) (n k + d) . The solution of problem (15) is determined by its Karush-Kuhn-Tucker (KKT) conditions [37], t (x - a) + G T m = 0, and Gx = h, where m ! R l is the Lagrangian multiplier. The KKT conditions imply a system of linear equations ta x C ; E = ; E, h m first quarter 2018 tI G T C =; G 0 E. t ||y - b||22 2 subject to y $ 0, minimize y x $ 0, n Based on (2), the linear system (16) can be efficiently mapped to memristor crossbars by configuring their memristance values according to the matrix C. On the other hand, problem (13) is equivalent to (16) (17) where b := x k +1 + (1/t) n k . The solution of problem (17) is determined by the projection of b onto the nonnegative orthant, y k +1 = (b) + . (18) Note that the positive part operator ($) + in (18) can be readily implemented using elementary logical or digital operations. We summarize the memristor-based LP solver in Fig. 4. Although LP is a relatively simple optimization problem, the LP solver illustrates our general idea and paves the way for numerous memristor-based applications in optimization problems. Our solution framework offers two major advantages. First, in the linear system (16), the coefficient matrix C is independent of the ADMM iteration so that memristors need to be configured only once. This feature makes it more attractive than gradient-type and interior-point algorithms, where memristors have to be reconfigured at each iteration [27]. Second, ADMM splits a complex problem into subproblems, each of which is easier to solve and implement in hardware. B. Quadratic Optimization With Memristors QP is an optimization problem whose objective and constraint functions involve quadratic and/or linear terms. There exist many variants of QP, such as a second-order cone program (SOCP) and a quadratically constrained quadratic program (QCQP) [37]. In this section, we focus on the design of a memristor-based solver for SOCP, Iterate Until Convergence Solve Linear System in (15) Using M Updated x k+1 M: Memristor Crossbars with Mapped C Project Onto Positive Orthant in (17) Using a Analog/Digital Technology Updated µk+1 Updated y k+1 Summing Amplifier Figure 4. memristor crossbar based solution framework in linear programming. ieee circuits and systems magazine 35

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