ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2018, Vol. 12, No. 2, pp. 325–333.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
E.V. Prosolupov, G.Sh. Tamasyan, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 2, pp. 82–100.
Complexity Estimation for an Algorithm of Searching for Zero
of a Piecewise Linear Convex Function
E. V. Prosolupov
and G. Sh. Tamasyan
St. Petersburg State University, Universitetskii pr. 35, St. Petersburg, 198504 Russia
Received March 10, 2017; in ﬁnal form, December 26, 2017
Abstract—It is known that the problem of the orthogonal projection of a point to the standard
simplex can be reduced to solution of a scalar equation. In this article, the complexity is analyzed of
an algorithm of searching for zero of a piecewise linear convex function which is proposed in . The
analysis is carried out of the best and worst cases of the input data for the algorithm. To this end, the
largest and smallest numbers of iterations of the algorithm are studied as functions of the size of the
input data. It is shown that, in the case of equality of elements of the input set, the algorithm performs
the smallest number of iterations. In the case of diﬀerent elements of the input set, the number of
iterations is maximal and depends rather weakly on the particular values of the elements of the set.
The results of numerical experiments with random input data of large dimension are presented.
Keywords: standard simplex, orthogonal projection of point, zeros of function
In many applied problems (astronomy, pattern recognition, mathematical diagnostics, and control
theory), as well as in the methods of mathematical programming (conditional and unconditional
optimization), it is required to ﬁnd the projection of a point to a set [2–5, 8, 11–16, 18, 19, 22–
25, 27, 32, 33]. Rather often the Euclidean norm is used; but, other norms can also be involved [6, 7]
(in the applications to economic problems).
It is known that the problem of the orthogonal projection of a point to a standard simplex is posed as
a problem of quadratic programming [8, 32]. In , for the ﬁrst time, the reduction of this problem to
solution of some scalar equation was presented. Later in , some algorithm was proposed for solving
a more general problem (of quadratic programming) which has linear order of complexity. We also note
the papers [9, 26, 29] in which the solution of this scalar equation is obtained by other methods.
An algorithm of searching for the projection of a point to a standard simplex proposed in  is based
on a diﬀerent principle. It has a clear geometrical interpretation that is noted in [10, 21]. The issue of
complexity of the algorithm from  is considered in [10, 17, 34].
In this article, the complexity is analyzed of the algorithm for ﬁnding a zero of the scalar equation
proposed in . At the level of ideas, it is close to . The analysis of the best and worst cases of
input data for the algorithm is carried out. To this end, the largest and smallest numbers of iterations
of the algorithm are under study as a function of the size of the input data. It is shown that, in the case
of equality of the elements of the input set, the algorithm performs the least number of iterations. In the
case of the input set of diﬀerent elements, the number of iterations is maximal and rather weakly depends
on the speciﬁc values of the elements of the set. The results of computational experiments with random
input data of high dimensionality are presented too.
The article is organized as follows: In Sections 1 and 2 the formal statement of the problem and the
description of the algorithm are given. Section 3 is devoted to analysis of complexity of the algorithm,
and Section 4 contains the results of numerical experiments.