Full Text
UDC 513.88.+519.3
MATHEMATICS
E. G. GOL'SHTEIN
DUAL PROBLEMS OF CONVEX AND FRACTIONAL-CONVEX PROGRAMMING IN FUNCTIONAL SPACES
(Presented by Academician L. V. Kantorovich on 31 VIII 1966)
The paper gives a general scheme for forming the dual problem for functional analogues of problems of convex and fractional-convex programming and formulates a number of assertions constituting the basis of the theory of duality for these problems. The proposed scheme for constructing the dual problem is analytic in character and generalizes the approach set forth in (1) for finite-dimensional problems of convex programming. We note that another, geometric principle for constructing the dual problem is contained in (2).
- Let \(E\) and \(E_1\) be real \(B\)-spaces; \(G\) and \(G_1\) nonempty convex subsets of \(E\) and \(E_1\), respectively; \(G_1\) a cone; \(f(x)\) a concave real functional defined on \(G\); \(\Phi(x)\) an operator from \(G\) into \(E_1\), concave (relative to \(G_1\)) on \(G\), i.e. satisfying the condition
\[
\Phi(\alpha x_1+(1-\alpha)x_2)-\alpha\Phi(x_1)-(1-\alpha)\Phi(x_2)\in G_1;
\]
\[
x_1,x_2\in G,\quad 0\leq \alpha\leq 1.
\]
We turn \(E_1\) into a partially ordered set by means of the condition
\[ y'\leq y''(G_1),\quad \text{if } y''-y'\in G_1\quad (y',y''\in E_1); \]
\(G_1\) is sometimes called the positive cone in \(E_1\). The positive cone \(G_1\) induces in \(E_1^*\) the convex cone \(G_1^*=\{\lambda:\lambda(y)\geq 0,\ \text{if } y\geq 0(G_1),\ \lambda\in E_1^*\}\), which is taken as the positive cone of this space.
The problem of maximizing the functional \(f(x)\):
\[ f(x)\to \sup \tag{1} \]
under the conditions
\[ \Phi(x)\geq 0\ (G_1), \tag{2} \]
\[ x\in G, \tag{3} \]
is naturally called a convex programming problem in a functional space. Let \(R=\{x:\Phi(x)\geq 0(G_1),\ x\in G\}\). We shall agree to call points \(x\in R\) plans of problem (1)—(3); a sequence \(X=\{x^{(k)}\}\) of plans \(x^{(k)}\) of problem (1)—(3), for which there exists
\[ \lim_{k\to\infty} f(x^{(k)})=f(X), \]
a plan-sequence of this problem. A plan-sequence \(X=\{x^{(k)}\}\) of problem (1)—(3) will be called a solution of the given problem if \(f(X)=v\), where
\[ v=\sup_{x\in R} f(x). \]
In particular, a stationary sequence \(\{x^{(k)}\}=\{x^*\}\) may turn out to be a solution of problem (1)—(3); the point \(x^*\) will be called a solution-plan of problem (1)—(3).
Put
\[ F(x,\lambda)=f(x)+\lambda(\Phi(x)),\quad x\in G,\quad \lambda\in E_1^*; \]
\(F(x,\lambda)\) is usually called the Lagrange functional of problem (1)—(3).
Let
\[ \varphi(x)=\inf_{\lambda\geqslant 0\ (G_1^*)} F(x,\lambda). \tag{4} \]
Consider the problem
\[ \varphi(x)\to \sup \tag{5} \]
under the condition
\[ x\in G. \tag{6} \]
It is not difficult to verify that, in the case when \(G_1\) is closed, problems (1)—(3) and (4)—(6) are equivalent (the concavity of \(f,\Phi\) and the convexity of \(G\) are not essential here). The structure of problem (4)—(6), which is equivalent to the original problem (1)—(3), suggests a natural formulation for the dual problem. Problem (4)—(6) consists in first applying to the Lagrange functional \(F(x,\lambda)\) the operation \(\inf\) with respect to \(\lambda\in G_1^*\), and then the operation \(\sup\) with respect to the elements \(x\in G\). In order to pass to the dual problem, we change the order in which these operations are applied.
Let
\[ \psi(\lambda)=\sup_{x\in G} F(x,\lambda). \tag{7} \]
We define the problem dual to (1)—(3) as follows: it is required to find
\[ \widetilde v=\inf \psi(\lambda) \tag{8} \]
under the condition
\[ \lambda\geqslant 0\ (G_1^*). \tag{9} \]
2. A sequence \(X=\{x^{(k)}\}\) of elements \(x^{(k)}\in G\) will be called a generalized plan of problem (1)—(3) if there exists a sequence \(\{y_1^{(k)}\}\) of elements \(y_1^{(k)}\geqslant 0\ (G_1)\) such that \(\Phi(x^{(k)})=y_1^{(k)}+y_2^{(k)}\), \(\lim_{k\to\infty}|y_2^{(k)}|=0\), and, moreover, there exists \(\lim_{k\to\infty} f(x^{(k)})=f(X)\).
Let \(\overline R\) be the set of generalized plans of problem (1)—(3). Obviously, if the generalized plan \(\{x^{(k)}\}=\{x^*\}\), then \(x^*\) is a plan of problem (1)—(3). Consequently, \(R\subset \overline R\).
By the generalized problem (1)—(3) we shall mean the problem of maximizing \(f(X)\) on the set \(\overline R\). Put \(v'=\sup_{X\in\overline R} f(X)\). We agree to regard \(v'=\infty\) (\(v=-\infty\)) if \(\overline R=\varnothing\) (\(R=\varnothing\)). The proposition formulated below, establishing a connection between the generalized problem (1)—(3) and the dual problem (7)—(9), is naturally called the generalized duality theorem.
Theorem 1. For an arbitrary convex programming problem (1)—(3), with \(\overline R\ne\varnothing\), the generalized duality relation holds:
\[ v'=\widetilde v. \tag{10} \]
As a consequence of Theorem 1 we obtain Theorem 2.
Theorem 2. When \(\overline R\ne\varnothing\), problem (1)—(3) and the problem dual to it (7)—(9) are connected by the duality relation:
\[ v=\widetilde v \tag{11} \]
if and only if
\[ v=v'. \tag{12} \]
In the case when the operator \(\Phi\) and the functional \(f\) are linear, and \(G\) is a cone, Theorem 1 (in a somewhat different formulation) was established in \((^3)\).
- With the aid of Theorem 2, the following two duality theorems are proved. Let
\[ M_\rho(f,G)= \begin{cases} \displaystyle \sup_{x\in G\cap C_\rho} f(x), & \text{if } G\cap C_\rho\ne \varnothing,\\ -\infty, & \text{if } G\cap C_\rho=\varnothing, \end{cases} \]
where \(C_\rho=\{x:\ |x|=\rho,\ x\in E\}\).
Theorem 3. Let (1)—(3) be a convex programming problem. If
\[ \lim_{\rho\to\infty} M_\rho(f,G)=-\infty; \tag{13} \]
\(G\) and \(G_1\) are closed sets; \(\Phi\) and \(f\) are continuous on \(G\); every bounded subset of \(G\) is weakly compact, then problems (1)—(3), (7)—(9) are connected by the duality relation (11).
Theorem 4. Let all the conditions listed in Theorem 3 be retained, with the exception of (13). If every bounded subset of \(G\) is compact, and the set of solution-plans of problem (1)—(3) is nonempty and bounded, then problems (1)—(3) and (7)—(9) are connected by the duality relation (11).
The conditions of Theorem 3 are satisfied, in particular, for a number of best-approximation problems with additional constraints. Theorem 4 finds application in the theory of finite-dimensional convex programming.
- The conditions of Theorems 3 (in the case \(R\ne \varnothing\)) and 4 obviously guarantee the existence of a solution-plan of problem (1)—(3). However, the dual problem (7)—(9), as the simplest finite-dimensional examples show, in general does not possess this property. The conditions listed in the theorem below not only guarantee that the duality relation is fulfilled, but also ensure the existence of a solution-plan of problem (7)—(9), provided only that \(\nu<\infty\). (At the same time, problem (1)—(3) may also have no solution-plan.) Let \(E=E_{01}\times E_{02}\), \(E_1=E_{11}\times E_{12}\), \(G=G_{01}\times G_{02}\), \(G_1=G_{11}\times G_{12}\), where \(E_{\gamma i}\) \((\gamma=0,1,\ i=1,2)\) is a \(B\)-space; \(G_{0i}\) is a convex subset of \(E_{0i}\); \(G_{1i}\in E_{1i}\) is a convex cone \((i=1,2)\).
Put
\[ f(x)=f(x_1,x_2);\qquad \Phi(x)=\bigl(\Phi_1(x_1,x_2),\ \Phi_2(x_1)+A(x_2)\bigr), \tag{14} \]
where \(x_i\in E_{0i}\) \((i=1,2)\); \(\Phi_1(x_1,x_2)\) is a concave (relative to \(G_{11}\)) operator acting from \(G\) into \(E_{11}\), \(\Phi_2(x_1)\) is a concave (relative to \(G_{12}\)) operator acting from \(G_{01}\) into \(E_{12}\); \(A(x_2)\) is a linear bounded operator from \(E_{02}\) into \(E_{12}\). We shall say that the constraints of problem (1)—(3), for \(f\) and \(\Phi\) defined by (14), satisfy the generalized Slater condition if there exists such a plan \(x^*=(x_1^*,x_2^*)\) of the problem that: a) \(\Phi_1(x_1^*,x_2^*)\) is an interior point of \(G_{11}\); b) \(x_2^*\) is an interior point of \(G_{02}\).
For \(E=E_{01}\), \(E_1=E_{11}\), the condition formulated becomes the familiar Slater condition \((^{4,5})\).
Theorem 5. Let (1)—(3), for \(f\) and \(\Phi\) defined by (14), be a convex programming problem, and let \(f(x_1,x_2)\) and \(\Phi(x_1,x_2)\) be continuous in \(x_2\in G_{02}\) for fixed \(x_1\in G_{01}\). If constraints (2), (3) satisfy the generalized Slater condition and the operator \(A\) maps \(E_{02}\) onto \(E_{12}\), then problems (1)—(3) and (7)—(9) are connected by the duality relation (11), and in the case \(\nu<\infty\) the lower bound (8) is attained.
A special case of Theorem 5 \((E=E_{01}, E_1=E_{11})\), in different terms and under somewhat stronger assumptions, is contained in \((^5)\).
- Closely connected with Theorems 3—5 are the so-called optimality criteria—necessary and sufficient conditions for a certain plan-sequence of a problem of type (1)—(3) to be its solution. Similar criteria were first studied by L. V. Kantorovich as applied to linear programming problems \((^6)\).
Theorem 6. Let \(X=\{x^{(k)}\}\) be a plan-sequence of problem (1)—(3), \(f(X)<\infty\). In order that \(X\) be a solution of problem (1)—(3), it is sufficient, and in the case when the duality relation (11) is satisfied also necessary, that there exist a sequence \(\{\lambda^{(k)}\}\), \(\lambda^{(k)}\geqslant 0\ (G_1^*)\), such that
\[ \lim_{k\to\infty} F(x^{(k)},\lambda^{(k)}) = \lim_{k\to\infty}\sup_{x\in G} F(x,\lambda^{(k)}), \qquad \lim_{k\to\infty}\lambda^{(k)}\bigl(\Phi(x^{(k)})\bigr)=0. \tag{15} \]
If, in addition to satisfaction of relation (11), the dual problem (7)—(9) has a plan-solution, then the sequence \(\{\lambda^{(k)}\}\) in Theorem 6 may be taken to be stationary, i.e. \(\{\lambda^{(k)}\}=\{\lambda^*\}\). Thus Theorems 3—5 single out classes of problems of the type (1)—(3) for which conditions (15) constitute an optimality criterion.
The formulations of the dual problems and optimality criteria admit refinements as applied to separate classes of problems of the type (1)—(3).
- Let \(\Phi_1(x)\) and \(\Phi_2(x)\) be operators from \(G\) into the \(B\)-space \(E_2\), and let \(G_2\) be a convex cone in \(E_2\). Consider the problem
\[ f(x)= \inf_{\mu\geqslant 0\,(G_2^*),\ |\mu|=1} \frac{\mu(\Phi_1(x))}{\mu(\Phi_2(x))} \to \sup \tag{16} \]
under conditions (2), (3).
It is natural to call the problem (16), (2), (3) a problem of fractional-convex programming in a functional space, if \(\Phi_1(x)\) is a concave operator with respect to \(G_2\); \(\Phi_2(x)\) is a concave (convex) operator with respect to \(G_2\) for \(v_1'>0\) (\(v_1'<0\)),
\[ v_1'=\sup_{x\in \overline{R}} f(X); \qquad \inf_{\mu\geqslant 0\,(G_2^*),\ |\mu|=1} \mu(\Phi_2(x))\geqslant \rho>0 \quad \text{for any } x\in G. \]
As the Lagrangian functional of problem (16), (2), (3) one takes
\[ F(x,\mu,\lambda) = [\mu(\Phi_1(x))+\lambda(\Phi(x))]/\mu(\Phi_2(x)). \]
Next put
\[ \psi(\mu,\lambda)=\sup_{x\in G} F(x,\mu,\lambda) \tag{17} \]
and define the problem dual to (16), (2), (3) as follows:
\[ \psi(\mu,\lambda)\to \inf, \tag{18} \]
\[ \mu\geqslant 0\ (G_2^*), \qquad |\mu|=1, \qquad \lambda\geqslant 0\ (G_1^*). \tag{19} \]
For the problem of fractional-convex programming (16), (2), (3) and the dual problem (17)—(19) to it, analogues of Theorems 1—6 hold; the formulations of the corresponding assertions for problems (1)—(3); (7)—(9) and (16), (2), (3); (17)—(19) differ only in insignificant details.
Central Economic-Mathematical Institute
Received
26 VIII 1966
CITED LITERATURE
- E. G. Gol’shtein, Economics and Mathematical Methods, 1, no. 3, 1965, pp. 410—424.
- G. Sh. Rubinshtein, DAN, 152, 288 (1963).
- R. J. Duffin, Linear Inequalities, IL, 1959, pp. 263—272.
- M. Slater, Cowles Commission Discussion Paper, Math., 403, Nov. (1950).
- L. Gurvits, in: Studies on Linear and Nonlinear Programming, IL, 1962, p. 65.
- L. V. Kantorovich, Mathematical Methods in the Organization and Planning of Production, L., 1939, p. 67.