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A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions
The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matrices and . Moreover, an interesting generalization of the geometric mean of and to convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.
The notion of geometric means is extended by Ando  to the case of positive semidefinite matrices and as the maximum of all for which is positive semidefinite. If is invertible, then . The geometric mean appears in the literature with many applications in matrix inequalities, semidefinite programming (scaling point [2, 3]), geometry (geodesic middle [4, 5]), statistical shape analysis (intrinsic mean [6, 7]), and symmetric matrix word equations [8–10]. The most important property of the geometric mean is that it has a Riccati matrix equation as the defining equation. The geometric mean is the unique positive definite solution of the Riccati matrix equation .
An interesting generalization of the geometric mean to convex functions was introduced by Atteia and Raïssouli  with a different viewpoint of the convex analysis. The natural idea to make an extension from positive semidefinite matrices to convex functions is nothing but the association of a positive semidefinite matrix with the quadratic convex function . Atteia and Raïssouli  provided a general algorithm to construct the (self-dual) geometric mean and the square root of convex functions. As pointed out in , self-dual operators are important in convex analysis and also arise in PDE.
The present work aims at providing a further development of the geometric mean of the convex functions mentioned above. We develop a new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” by exploiting the proximal average of convex functions by Bauschke et al.  and investigate its essential properties such as limiting behaviors, self-duality, and monotonicity with respect to parameters. While doing so, we will see that the geometric mean due to Atteia and Raïssouli  can be interpreted as an element of “the geometric mean of parameterized arithmetic and harmonic means of convex functions” with the particular parameter .
In fact, this work is motivated by a recent result due to Kim et al.  concerned with a new matrix mean. Actually, the geometric mean of parameterized arithmetic and harmonic means of convex functions is an extension of the new matrix mean to a convex function mean under a standard setting with two convex functions.
2. Geometric Mean and -Mean of Parameter
We begin with the algorithm of finding the geometric mean of two proper convex lower semicontinuous functions and introduced by Atteia and Raïssouli [11, Proposition 4.4] and some comments on the procedure. Let with where denotes the class of proper convex lower semicontinuous functions from the Euclidean space to . Set two sequences of convex functions and recursively: where stands for the Fenchel conjugate of .
It is claimed that all the and do belong to [11, Proposition 4.4]. However, to ensure this property, we need more. Indeed, we see where stands for the infimal convolution. As is well known, can take as a value so it may not be proper. This happens for two simple linear functionals and in the one-dimensional case. So the properness of equivalent to that of is not safe. Exactly the same problem may occur whenever is defined. Moreover, it is not sure that is proper because can be empty. Thus the basic necessity that and belong to is not guaranteed under the general assumption only that with in . Hence it is necessary to impose a suitable condition to meet this demand. For that purpose, recall that a function is called cofinite if the recession function of satisfies , for all (see [15, page 116]). Then is cofinite if and only if by means of [15, Corollary ]. The terminology “cofinite” is renewed as “coercive” in [16, 3.26 Theorem].
Now we take a look at Atteia and Raïssouli [11, Proposition 4.4] with a refined proof.
Proposition 2.1 (See Atteia and Raïssouli [11, Proposition 4.4]). Let . If either or is cofinite, then all and belong to and is cofinite for all . Hence the geometric mean due to Atteia and Raïssouli , that is, the limit is well defined and proper convex on . In particular, it belongs to under the assumption that either or is closed. Moreover, under the condition .
Proof. Without loss of generality, we may assume that is cofinite. Clearly, since . In addition, is still cofinite by [15, Theorem 9.3]. Then by virtue of [15, Corollary ]. Thus . By induction, assume that
Then , so . Moreover, is cofinite because is cofinite. It is readily checked that
Hence . In this case, . Thus we obtain that
According to Atteia and Raïssouli [11, Proposition 4.4], we have
Hence the geometric mean is well defined and belongs to under the given hypothesis. (If is closed, we define an increasing sequence by
where denotes the indicator function of the closed convex set . Obviously, is the common limit of and , hence, belongs to .)
For the equality , we have Hence On the other hand, Thus Therefore we get
Remark 2.2. The well definedness of is readily checked by the assumption is cofinite. (Without this condition, may not be well defined so that the identity breaks down.) With the additional property that is closed, we have . Hence
Proposition 2.1 provides a sufficient condition to entail the validity of [11, Proposition 4.4]. It is also mentioned in [11, Remark 4.5] that if and are finite-valued, is satisfied. But even though it is true, can be identically as shown in the case of and in so that the limiting process using (2.7) may not be available any more. So some restrictions should be imposed to properly define the geometric mean of two convex functions and . Of course, for an , the geometric mean and the convex square root of (see [11, Definition 4.7]) are always well defined because is cofinite. What is a minimal assumption? That is a question to be answered.
Throughout this paper, we adopt the following modified definition of proximal average for the convenience of presentation. For , with , where , , each belongs to , and ’s are positive real numbers with .
From now on, we consider the simple case where , , and with . Define two sequences of convex functions and recursively as follows:
Theorem 2.3. For , one has(i) and , for all ;(ii) and , for all ;(iii), for all ;(iv)there exists a limit which is a proper convex function with . Furthermore, if either or is closed, is the common limit of and for some increasing sequence . In this case, .
Proof. (i) Since , by Bauschke et al. [13, Theorem 4.6],
because is a convex set. By induction, assume that . Then
Thus we obtain that
This implies that, for all , and with the help of [13, Corollary 5.2].
(ii) The first assertion is a direct consequence of [13, Theorem 5.4]. For the second, by definition and the first assertion, we see For the last, observe that which is nothing but the first assertion. Note that all the arithmetics are safe because both and are finite-valued.
(iii) By (ii) and the extended arithmetic (see ), we get
(iv) From (ii), we have Hence if by (2.19), converges to a real number . If , . Let the limit function be . Clearly, is proper convex because is convex. Moreover, if , by (iii) and (2.23), it is the common limit of and , so since it is a supremum of . If is closed, we define an increasing sequence by where denotes the indicator function of the closed convex set . Obviously, is the common limit of and , hence belongs to .
Remark 2.4. If both and are finite-valued, the condition is automatically satisfied.
Corollary 2.5. For and with ,(i),(ii).
Now we express in terms of a geometric mean.
Theorem 2.6. Let . For with , one has
Proof. Claim 1. We have
Indeed, put and . Then because and are finite-valued, and is cofinite by [15, Theorem 9.3]. By Proposition 2.1, we obtain
where and are defined as in (2.1). Set, for each ,
Then by (2.5)
Put and . Also define
Then we have
Moreover, it follows from (2.30) that and satisfy the recursion formula in (2.1). From Theorem 2.3 and (2.28), we get
Claim . .
Set two cofinite functions and . It sufficies to check that In fact, let and . Then and belong to , and is cofinite by Proposition 2.1. Clearly, we have Again appealing to (2.6) yields that This completes the proof.
Now we give the following name to by Theorem 2.6 above.
Definition 2.7. For , one defines This is called the geometric mean of parameterized arithmetic and harmonic means of and and abbreviated by “-mean of parameter ”.
3. Properties of -Mean of Parameter
To deal with (for all ), in what follows, we assume the following for the simplicity of arguments.
3.1. Constraint Qualifications
Consider with ,, either is cofinite and is closed or is cofinite and is closed.
With these hypotheses, for all , is well-defined and is in .
Theorem 3.1. One has the limiting property:
Proof. For , by Corollary 2.5, we get By Bauschke et al. [13, Theorem 8.5], Thus Again appealing to Corollary 2.5 yields that By the self-duality of the proximal average [13, Theorem 5.1], we have Taking the limit in (3.5), we see from (3.6) that where the equality comes from [13, Theorem 8.5]. By , ; hence we have Therefore it follows from (3.7) and (3.8) that This completes the proof.
Theorem 3.2. One has
(i) , for ,
(ii) (self-duality) , for all .
Proof. (i) According to Corollary 2.5 (ii), for . For , by Definition 2.7.
(ii) If , by definition, , so because . If , then by virtue of Proposition 2.1 and Remark 2.2. Let . Then by definition, , as desired.
Proposition 3.3. Let and for each . Then, for , where , and ’s are positive real numbers with .
Proof. For each , clearly
Theorem 3.4 (monotonicity). One has, for ,
Proof. Let . Clearly by [13, Theorem 8.5]. To use induction, assume that Then by (3.14), Proposition 3.3, and [13, Theorem 8.5]. Thus (3.14) holds for all . Hence, we get On the other hand, for , by means of (3.16). Now let . Recall that and (see (2.16), (2.1), and Corollary 2.5 (ii)). Assume that Then by virtue of (3.18), Proposition 3.3, [13, Theorem 5.4], and (2.5). Hence (3.18) holds for all . This implies that So, we get by (3.20) and Proposition 2.1. Therefore, the result follows from (3.16), (3.17), (3.20), (3.21), and Theorem 3.1.
Corollary 3.5. Let and be two (symmetric) positive definite matrices. Then, for , one has where Here denotes the matrix geometric mean of two positive definite matrices.
Proof. For a positive definite matrix , define the convex quadratic function Put and , then and clearly satisfy the constraint qualifications –. Applying Theorem 2.6 to these functions yields that where the second equality comes from Atteia and Raïssouli [11, Proposition 3.5 (v) and (vii)]. Since by Theorem 3.4, we have
Remark 3.6. Corollary 3.5 is a particular case of Kim et al. [14, Theorem 3.6] and is based on a different proof using a convex analytic technique in the case of two variables with no weights. To prove the monotonicity of w.r.t. the parameter , Kim et al.  exploited a well-known variational characterization of the geometric mean of two positive definite matrices.
We close this section with one more observation.
Definition 3.7 (See Bauschke et al. [13, Definition 9.1]). Let and be functions from to . Then epiconverges to , in symbols, , if the following hold for every :
(ii) and ,
The epitopology is the topology induced by epiconvergence.
Proposition 3.8. One has
The first author was supported by the Basic Science Research Program through the NRF Grant no. 2012-0001740.
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1970.View at: MathSciNet