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CYBERNETICS AND CONTROL THEORY
M. V. RYBASHOV, E. E. DUDNIKOV
A PARAMETRIC METHOD FOR SOLVING FRACTIONAL PROGRAMMING PROBLEMS ON ANALOG COMPUTERS
(Presented by Academician V. A. Trapeznikov on 18 XI 1964)
1°. Consider the nonlinear programming problem
\[ \lambda^*=\min\{P(x),\, x\in R\} \tag{1} \]
with fractional objective function \(P(x)=F_1(x)/F_2(x)\) and a nonempty, admissible, bounded and closed set \(R\), defined by the system of constraints
\[ \varphi_s(x)\leq 0,\qquad s=1,\ldots,m, \tag{2} \]
\[ \psi_j(x)=\sum_{k=1}^{n} a_{jk}x_k+b_j=0,\qquad j=1,\ldots,q;\ q<n, \]
\[ \alpha_i\leq x_i\leq \beta_i,\qquad i=1,\ldots,n, \]
where \(x\in E_n\); \(F_1(x), \varphi_s(x)\) are convex functions; \(a_{jk}, b_j, \alpha_i, \beta_i\) are real numbers.
It is assumed that a segment of values of the parameter numbers \(\Omega=[\lambda_1,\lambda_2]\) is known, among which lies the extremal value (1), \(\lambda^*\in\Omega\). Further, it is assumed that \(F_2(x)>0\) for \(x\in R\), and that the auxiliary function \(\xi(x,\lambda)=-\lambda F_2(x)\), \(\lambda\in\Omega\), is convex.
In addition, it is assumed that the functions \(F_1(x), F_2(x), \varphi_s(x)\) are defined everywhere, continuously differentiable, and that their partial derivatives in every bounded closed domain satisfy a Lipschitz condition. The fractional programming problem is: find a vector \(x=x^*\), \(x^*\in R\), for which the objective function \(P(x)\) attains the extremal value \(\lambda^*\).
Remark 1. If \(F_2(x)=1\) (convex programming) or is linear, then there is no need to know the segment of numbers \(\Omega\). In this case the function \(\xi(x,\lambda)\) is convex for any \(\lambda\).
Remark 2. The problem formulated includes the known problem of fractional-linear programming \(\left(^{1,2}\right)\) under additional constraints \((2')\).
2°. Introduce the function \(V(x,\lambda)\) with parameter \(\lambda\) (\(\lambda\) is a real number)
\[ \begin{aligned} V(x,\lambda)=&\ \frac{1}{2}\{[F_1(x)-\lambda F_2(x)]^2 \operatorname{sg}[F_1(x)-\lambda F_2(x)] \\ &+\sum_{s=1}^{m}\varphi_s^2(x)\operatorname{sg}\varphi_s(x) +\sum_{j=1}^{q}\psi^2(x) +\sum_{i=1}^{n}[(x_i-\beta_i)^2\operatorname{sg}(x_i-\beta_i) \\ &\qquad\qquad\qquad\qquad +(\alpha_i-x_i)\operatorname{sg}(\alpha_i-x_i)]\}, \end{aligned} \]
where \(\operatorname{sg} r=1\) for \(r>0\) and \(\operatorname{sg} r=0\) for \(r\leq 0\). Let \(\Omega'\) be the set of values of the function \(P(x)\) on the set \(R\), \(\lambda^*\in\Omega'\) and \(\Omega\cap\Omega'\neq 0\).
The function \(V(x,\lambda)\) is everywhere convex and attains an absolute minimum on the compact convex set \(Q(\lambda)\), moreover
\[ V(x,\lambda)=0\quad \text{for } x\in Q(\lambda),\ Q(\lambda)\subseteq R,\ \lambda\in\Omega', \tag{3} \]
\[ V(x,\lambda)>0\quad \text{in all other cases.} \]
From the convexity of \(V(x,\lambda)\) and the boundedness of the set \(Q(\lambda)\) it follows that
\[ \lim_{\|x\|\to+\infty} V(x,\lambda)=+\infty , \tag{4} \]
\[ \|\operatorname{grad} V(x,\lambda)\|>0 \quad \text{when } V(x,\lambda)>0 . \tag{5} \]
If \(\lambda=\lambda^*\), then the set \(Q(\lambda^*)\) consists precisely of the points at which the extremal value \(P(x)=\lambda^*\) is attained.
Using the properties (3) of the function \(V(x,\lambda)\), the fractional-programming problem can be reduced to finding the minimum value of the parameter \(\lambda\in\Omega\) and the corresponding vector \(x\) for which \(V(x,\lambda)=0\).
The procedure for finding \(\lambda=\lambda^*\) and \(x=x^*\) reduces to the following. For \(\lambda=\lambda_1\) the function \(V(x,\lambda_1)\) is minimized. If \(\min_x V(x,\lambda_1)=0\), then \(\lambda_1=\lambda^*\). If, however, \(\min_x V(x,\lambda)>0\), then the minimization is repeated for an increasing sequence of values of the parameter \(\lambda\) until such a \(\lambda_i\) is found for which \(\min_x V(x,\lambda_i)=0\) and \(\min_x V(x,\lambda_i-\varepsilon)>0\) for any arbitrarily small \(\varepsilon>0\). In this case \(x=x^*\), \(\lambda_i=\lambda^*\).
3°. To find the minimum of \(V(x,\lambda)\) for fixed \(\lambda\), we use the gradient system of differential equations
\[ \tau_i \frac{dx_i}{dt} = -\frac{\partial V(x,\lambda)}{\partial x_i}, \qquad \tau_i>0, \qquad i=1,\ldots,n. \tag{6} \]
The solutions of this system \(x(x_0,\lambda,t)\), independently of the initial conditions \(x_0\), \(x_0\in Q(\lambda)\), for \(t>0\) and \(t\to+\infty\) converge to the set \(Q(\lambda)\), i.e., for any \(\delta\)-neighborhood of the set \(Q(\lambda)\) there exists a \(T(\delta,x^0)>0\) such that, for \(t>T(\delta,x^0)\), \(x(x_0,\lambda,t)\in\delta\) and \(\|x(x_0,\lambda,t)\|<M,\ M>0\).
Indeed, by virtue of (5) and (6),
\[ \frac{dV(x,\lambda)}{dt} = \sum_{i=1}^{n} \frac{\partial V(x,\lambda)}{\partial x_i} \frac{dx_i}{dt} \le -\frac{1}{\max_i\{\tau_i\}} \|\operatorname{grad} V(x,\lambda)\|^2 <0 \quad \text{when } x\notin Q(\lambda), \]
\[ dV(x,\lambda)/dt \equiv 0 \quad \text{when } x\in Q(\lambda). \]
It follows from (4) that every solution \(x(x^0,\lambda,t)\) is bounded. The set \(Q(\lambda)\) is an invariant set for the system (6), \(dV(x(x_0,\lambda,t),\lambda)/dt\equiv0\), if \(x^0\in Q(\lambda)\) and \(x\in Q(\lambda)\) for \(0\le t<+\infty\), and is an \(\omega\)-limit set for its solutions, which proves assertion (3). In this case the function \(V(x,\lambda)\) is an analogue of the Lyapunov function.
Taking (2) and \((2')\) into account, system (6) takes the form
\[ -\tau_i \frac{dx_i}{dt} = \left\{ \left[ \frac{\partial F_1(x)}{\partial x_i} - \lambda \frac{\partial F_2(x)}{\partial x_i} \right] \left[ F_1(x)-\lambda F_2(x) \right] \operatorname{sg}\left[ F_1(x)-\lambda F_2(x) \right] + \right. \]
\[ \left. + \sum_{s=1}^{m} \frac{\partial \varphi_s(x)}{\partial x_i} \varphi_s(x)\operatorname{sg}\varphi_s(x) + \sum_{j=1}^{q} \frac{\partial \psi_j(x)}{\partial x_i} \psi_j(x) + \right. \]
\[ \left. + \sum_{i=1}^{n} \left[ (x_i-\beta_i)\operatorname{sg}(x_i-\beta_i) - (\alpha_i-x_i)\operatorname{sg}(\alpha_i-x_i) \right] \right\}. \tag{7} \]
The system of differential equations (7) can be solved by known methods on an electronic analog computer (electronic model) [4]. With a sufficiently slow change of the parameter \(\lambda\), the analog model will automatically track, with some error \(\mu\), the coordinates of the minimum of the function \(W(x,\lambda)\), with \(\mu\to0\) as the rate of change of \(\lambda\) is decreased.
At the moment when the minimum of the function \(V(x,\lambda)\) passes from a nonzero value to a zero value, the vector \(x\) and the parameter \(\lambda\) are recorded. The obtained values \(x,\lambda\) are the solution of the problem.
Institute of Automation and Telemechanics
(Technical Cybernetics)
Received
11 XI 1964
CITED LITERATURE
- I. R. Isbell, W. H. Marlow, Nav. Res. Logist. Quart., 3, No. 1–2, 71 (1956).
- A. Charnes, W. W. Cooper, Nav. Res. Logist. Quart., 9, No. 3–4, 181 (1962).
- J. La-Salle, S. Lefschetz, Stability by Liapunov’s Direct Method, 1964.
- B. Ya. Kogan, Electronic Modeling Devices and Their Application to the Study of Automatic Control, Moscow, 1962.