Circular cone convexity and some inequalities associated with circular cones
© Zhou et al.; licensee Springer. 2013
Received: 17 July 2013
Accepted: 13 November 2013
Published: 4 December 2013
The study of this paper consists of two aspects. One is characterizing the so-called circular cone convexity of f by exploiting the second-order differentiability of ; the other is introducing the concepts of determinant and trace associated with circular cone and establishing their basic inequalities. These results show the essential role played by the angle θ, which gives us a new insight when looking into properties about circular cone.
MSC:26A27, 26B05, 26B35, 49J52, 90C33, 65K05.
Real applications of a circular cone lie in some engineering problems, for example, in the formulation for optimal grasping manipulation for multi-fingered robots, the grasping force of i th finger is subject to a circular cone constraint, see [10, 11] and references for more details.
with if , and being any vector w in satisfying if . Clearly, if and only if .
Throughout this paper, we always assume that J is an interval in ℝ. Clearly, as , reduces to the second-order cone and the above expressions (1) and (2) correspond to the spectral decomposition and the SOC-function associated with the second-order cone, respectively (see [13, 14] for more information regarding ).
It is well known that in dealing with symmetric cone optimization problems, such as second-order cone optimization problems and positive semi-definite optimization problems, this type of vector-valued functions plays an essential role. Inspired by this, we study the properties of , which is crucial for circular cone optimization problems. In our previous works, we have studied the smooth and nonsmooth analysis of [8, 10]; and the circular cone monotonicity and second-order differentiability of . From the aforementioned research, there is an interesting observation: some properties commonly shared by and are independent of the angle θ; for example, is directionally differentiable, Fréchet differentiable, semi-smooth if and only if f is directionally differentiable, Fréchet differentiable, semi-smooth; while some properties are dependent on the angle θ; for example, with for is circular cone monotone as , but not circular cone monotone as .
The characterization of -convexity is based on the observation that f is -convex if and only if for all . Our result shows that the circular cone convexity requires that the angle θ belongs in . In particular, we show that f is -convex of order 2 if and only if and f is convex.
where , fails as but holds as ; (iii) the third class always fails no matter what value of θ is chosen. These results give us a new insight into a circular cone and make us focus more on the role played by the angle θ.
The notation used in this paper is standard. For example, denote by the n-dimensional Euclidean space and by the set of all nonnegative real scalars, i.e., . For , the inner product is denoted by . Let mean the spaces of all real symmetric matrices in , and let denote the cone of positive semi-definite matrices. We write to stand for . Finally, we define for convenience.
2 Circular cone convexity
The main purpose of this section is to provide characterizations of -convex functions. First, we need the following technical lemma.
then for all .
Proof If , then . From , we have . Thus, by letting and by letting .
If , then . From , we obtain and .
which is ensured by condition (4) and the fact since . This completes the proof. □
Lemma 2.2 [, Theorem 3.1]
then f is -convex. Here for .
Proof According to [, Theorem 3.2], f is -convex if and only if for all and . We proceed the proof by considering the following three cases.
Hence, if and only if .
Dividing by both sides and taking the limits as , we obtain . Since can take an arbitrary value in J, it is clear that (14) is equivalent to saying that for all , i.e., f is convex on J. Indeed, the condition is ensured by the fact that for some .
In view of Lemma 2.1, the condition means , is ensured by the convexity of f (see (14)), corresponds to (6), and corresponds to (7). In addition, condition (4) takes the special form (9) and (10), respectively. □
Theorem 2.2 Suppose that is second-order continuously differentiable. Then f is -convex of order 2 on S if and only if and f is convex on J.
This is ensued by the conditions that and f is convex on J. Thus, the proof is complete. □
If, in particular, , then (6) and (7) reduce to [, (21) in Proposition 4.2]; (9) reduces to [, (22) in Proposition 4.2]. In addition, due to (7), (8) holds automatically in this case. The above results indicate that the -convexity is dependent on the properties of f and the angle θ together.
3 Inequalities associated with circular cone
In this section, we establish some inequalities associated with circular cone, which we believe will be useful for further analyzing the properties of and proving the convergence of interior point methods for optimization problems involved in circular cones.
If , then , , and for ,
and for all .
In the following, we show that, in the framework of circular cone, the above inequalities can be classified into three categories. The first class holds independent of the angle, e.g., (a); the second class holds dependent on the angle, e.g., (b)-(e); the third class fails no matter what value of the angle is chosen, e.g., (f).
for all ;
If , then .
- (b)Since , we know
i.e., . □
- (a)For all ,
- (b)For all and ,
- (c)If and , then(20)
- (d)If and , then(21)
which is the desired result.
The result follows from the fact that for all .
- (c)Since , . For , there exist two nonnegative scalars such that and . This implies
- (d)For , since and , we know
Inequality (19) fails when . For example, let , , and . Then and , which says .
Inequality (20) fails when . For example, let , , and . Then .
Inequality (21) fails when . For example, for , , and . Then , , , and .
The precise relationship between and is provided as below.
Note that is positively homogeneous, i.e., for all . This together with Theorem 3.3 yields the following result.
Corollary 3.1 The trace is concave as and is convex as .
These results further indicate that the angle plays an essential role for a circular cone. As in symmetric cone optimization, we believe that these inequalities about and are key ingredients in penalty and barrier functions which can be adapted in designing barrier and penalty algorithms (including interior point algorithm) for circular cone optimization. This merits our further research.
The second author is a member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.
We are gratefully indebted to anonymous referees for their valuable suggestions that helped us to essentially improve the original presentation of the paper. The first author’s work is supported by the National Natural Science Foundation of China (11101248, 11271233) and Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2012AM016). The second author’s work is supported by the National Science Council of Taiwan.
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