DOI 10.1007/s10898-007-9156-y O R I G I NA L PA P E R

**Entropy-like proximal algorithms based** **on a second-order homogeneous distance** **function for quasi-convex programming**

**Shaohua Pan** **· Jein-Shan Chen**

Received: 13 August 2006 / Accepted: 12 March 2007 / Published online: 25 April 2007

© Springer Science+Business Media LLC 2007

**Abstract** We consider two classes of proximal-like algorithms for minimizing a proper
*lower semicontinuous quasi-convex function f(x) subject to non-negative constraints x ≥ 0.*

The algorithms are based on an entropy-like second-order homogeneous distance function.

Under the assumption that the global minimizer set is nonempty and bounded, we prove the
full convergence of the sequence generated by the algorithms, and furthermore, obtain two
important convergence results through imposing certain conditions on the proximal param-
eters. One is that the sequence generated will converge to a stationary point if the proximal
parameters are bounded and the problem is continuously differentiable, and the other is that
the sequence generated will converge to a solution of the problem if the proximal parameters
approach to zero. Numerical experiments are done for a class of quasi-convex optimization
*problems where the function f(x) is a composition of a quadratic convex function from IR** ^{n}*
to IR and a continuously differentiable increasing function from IR to IR, and computational
results indicate that these algorithms are very promising in finding a global optimal solution
to these quasi-convex problems.

**Keywords** Proximal-like method· Entropy-like distance · Quasi-convex programming

Shaohua Pan work was partially supported by the Doctoral Starting-up Foundation (05300161) of GuangDong Province.

Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. Jein-Shan Chen work is partially supported by National Science Council of Taiwan.

S. Pan (

### B

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School of Mathematical Sciences, South China University of Technology, Guangzhou 510641, China e-mail: shhpan@scut.edu.cn

J.-S. Chen

Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan e-mail: jschen@math.ntnu.edu.tw

**1 Introduction**

*The proximal point algorithm for minimizing a convex function f(x) on IR** ^{n}* generates a
sequence

*{x*

*}*

^{k}*k∈N*⊆ IR

*by the following iterative scheme:*

^{n}*x** ^{k+1}*= argmin

*x∈IR*^{n}

*f(x) + λ**k**x − x*^{k}^{2}

*,* (1)

where*λ**k*is a sequence of positive numbers and · denotes the Euclidean norm in IR* ^{n}*. This
method, originally introduced by Martinet [15], is based on the Moreau proximal approxi-

*mation of f (see [16]). The proximal point algorithm was then further developed and studied*by Rockafellar [19,20]. Later, several researchers [4,5,7,12,14,23] proposed and studied nonquadratic proximal point algorithm by replacing the quadratic distance in (1) with a Bregman distance or an entropy-like distance. Among others, the entropy-like distance, also called

*ϕ-divergence, is defined by*

*d*_{ϕ}*(x, y) =*

*n*
*i*=1

*y**i**ϕ(x**i**/y**i**),* (2)

where*ϕ: IR → (−∞, +∞] is a closed proper strictly convex function satisfying certain*
conditions; see [12,13,23,24]. This class of distance-like functions was first proposed by
Teboulle [23] in order to define entropy-like proximal maps. A popular choice of*ϕ is the*
case that*ϕ(t) = t ln t − t + 1, for which the corresponding d** _{ϕ}*is exactly the well-known
Kullback–Leibler entropy function from statistics [7,8,10,23] and that is the “entropy" ter-
minology stems from.

The proximal-like algorithm based on*ϕ-divergence, originally designed for minimizing*
*a convex function f(x) subject to non-negative constraints x ≥ 0, consists of a sequence*
*{x** ^{k}*}

*k*

*∈N*⊆ IR

^{n}_{++}generated by the iterative scheme as follows:

*x*^{0}*> 0,*
*x** ^{k+1}*= argmin

*x≥0* *{ f (x) + λ**k**d*_{ϕ}*(x, x*^{k}*)}.* (3)
This class of proximal-like algorithms were studied extensively for convex programming;

see [12,13,23,24] and references therein, and particularly, the one with*ϕ(t) = t ln t − t + 1*
was recently extended to convex semidefinite programs [6] and convex second-order cone
programs in a recent manuscript of J.-S. Chen. In fact, the algorithm (3) associated with
*ϕ(t) = − ln t + t − 1 was first proposed by Eggermont [*8]. It is worth to point out that the
fundamental difference between (1) and (3) is that the term d_{ϕ}*(·, ·) is used in (*3) to force the
iterates*{x** ^{k}*}

*k∈N*to stay in IR

_{++}

*which is the interior of the non-negative orthant, namely the algorithm (3) will automatically generate a positive sequence*

^{n}*{x*

*}*

^{k}*k*

*∈N*⊆ IR

^{n}_{++}.

In this paper, we will focus on two classes of proximal-like algorithms of the form (3) but
*with a second-order homogeneous distance-like function d** _{φ}*given by

*d*_{φ}*(x, y) =*

*n*
*i*=1

*y*_{i}^{2}*φ(x**i**/y**i**),* (4)

where the kernel*φ is defined with two types of special ϕ and a quadratic function. The*
definition of*φ and the properties of d** _{φ}*are given in Sect. 3. This class of algorithms has been
studied for convex minimization (see [1,2,22]). However, we in this paper employ these

algorithms to solve the following quasi-convex minimization problem:

*min f(x)*

s*.t. x ≥ 0,* (5)

*where f*: IR* ^{n}* → IR is a proper lower semicontinuous quasi-convex function. Since we do

*not require the convexity of f , the basic iterative scheme for the algorithms is as follows:*

*x*^{0}*> 0,*
*x** ^{k+1}*∈ argmin

*x*≥0 *{ f (x) + λ**k**d*_{φ}*(x, x*^{k}*)},* (6)
where*λ**k*is same as before. The purpose of this paper is to establish the full convergence of
the sequence*{x** ^{k}*}

*k∈N*generated by (6) under some mild assumptions for the quasi-convex problem (5), and verify the effectiveness of the algorithms by numerical experiments.

Note that (5) is a special nonconvex optimization problem, and therefore the global opti- mization methods [11] developed for the general nonconvex optimization problem can be applied for solving it. Nevertheless, we should point out that the design of these global optimization methods is often far more complex than that of the proximal-like method (6).

The rest of this paper is organized as follows. In Sect.2, we recall some definitions and
basic results that will be used in the later sections. In Sect.3, we present the definition of the
kernel*φ and investigate the properties of d**φ*. Based on the entropy-like second-order homo-
*geneous distance function d** _{φ}*, we in Sect.4propose two classes of proximal-like algorithms,
and prove the full convergence of the sequence generated. In Sect.5, numerical experi-

*ments were done with a specific d*

*for a class of continuously differentiable quasi-convex programming problems.*

_{φ}Unless otherwise stated, in this paper, we use the notation*·, · and · to denote the*
Euclidean inner product and Euclidean norm in IR* ^{n}*, and IR

_{+}

*to represent the non-negative orthant in IR*

^{n}*with the interior IR*

^{n}

^{n}_{++}

*. For a given differentiable function f*: IR

^{n}*→ IR, ∇ f (x)*

*denotes the gradient of f at x, while(∇ f (x))*

*i*

*means the i th partial derivative of f with*

*respect to x. In addition, we use*∇1

*d*

_{φ}*(x, y) to denote the partial derivative of d*

*with respect to its first component.*

_{φ}**2 Basic concepts**

In this section, we recall some definitions and basic results which will be used in the subse- quent analysis. We start with the definition of Fejér convergence for a sequence.

**Definition 2.1 A sequence***{y** ^{k}*}

*k∈N*

*is Fejér convergent to a nonempty set U*⊆ IR

*with*

^{n}*respect to a distance-like function d(·, ·), if for every u ∈ U, we have d(u, y*

^{k+1}*) ≤ d(u, y*

^{k}*).*

*When d is the Euclidean distance,{y*^{k}*} is called Fejér convergent to U.*

*Given an extended real-valued function f*: IR* ^{n}* → IR ∪ {+∞}, denote its domain by

*dom f:= {x ∈ IR*

^{n}*: f (x) < +∞}*

and its epigraph by

*epi f*:=

*(x, β) ∈ IR*^{n}*× IR : f (x) ≤ β*
*.*

*Then, f is said to be proper if dom f* *= ∅ and f (x) > −∞ for any x ∈ dom f , and f is*
*a lower semicontinuous function if epi f is a closed subset of IR** ^{n}*× IR. We next recall the
definition of the Fréchet subdifferential; see [18, Chapter 8] and [21, Chapter10].

* Definition 2.2 Let f*: IR

*→ IR ∪ {+∞} be a proper lower semicontinuous function. For*

^{n}*each x∈ dom f , the Fréchet subdifferential of f at x, denoted by ˆ∂ f (x), is the set of vectors*

*s*∈ IR

*such that*

^{n}lim inf

*y**=x,y→x*

1

*y − x*

*f(y) − f (x) − s, y − x*

*≥ 0.* (7)

*If x* */∈ dom f , then ˆ∂ f (x) = ∅.*

*The vector s satisfying the inequality (7) is also termed as a regular subgradient of f at x*
(see [21, p. 301]). It is not difficult to see that the inequality (7) is equivalent to

*f(y) ≥ f (x) + s, y − x + o(y − x),*
where

*y→x*lim*o(y − x)/y − x = 0.*

For the subdifferential ˆ*∂ f (x), the following results hold by direct verifications.*

**Lemma 2.3 [21, Chapter 8] Let f**: IR^{n}*→ IR ∪ {+∞} be a proper lower semicontinuous*
*function and ˆ∂ f (x) be the subdifferential of f at x. Then,*

(a) ˆ*∂ f (x) is a closed and convex set.*

*(b) If f is differentiable at x or in a neighborhood of x, then ˆ∂ f (x) = {∇ f (x)}, where*

*∇ f (x) is the gradient of f .*

*(c) If g* *= f + h with f finite at x and h differentiable on a neighborhood of x, then*
*ˆ∂g(x) = ˆ∂ f (x) + ∇h(x).*

*(d) If f has a local minimum at* *¯x, then 0 ∈ ˆ∂ f ( ¯x).*

To work with differentiable minimization problems, we also need the following definition.

* Definition 2.4 Suppose that f*: IR

*→ IR is a differentiable function. Then,*

^{n}*(a) For an unconstrained optimization problem of minimizing f(x) over x ∈ IR*^{n}*, x*^{∗}is
called a stationary point if*∇ f (x*^{∗}*) = 0.*

*(b) For a constrained optimization problem of minimizing f(x) over x ∈ C where C is*
nonempty and convex subset of IR^{n}*, x*^{∗}is called a stationary point if

*∇ f (x*^{∗}*)*^{T}*(x − x*^{∗}*) ≥ 0 for all x ∈ C.*

To close this section, we recall the concept of quasi-convexity, strict quasi-convexity and strong quasi-convexity, and briefly discuss general properties of the minimization problem involving the objective function with such properties.

* Definition 2.5 Let f*: IR

^{n}*→ IR be a proper function. Then, f is called quasi-convex if for*

*all x, y ∈ dom f and β ∈ (0, 1), there always holds*

*f(βx + (1 − β)y) ≤ max{ f (x), f (y)}.*

It can be proved that any convex function is also quasi-convex, but the converse is not true.

For a quasi-convex function, we have the following important property.

* Proposition 2.6 The proper function f*: IR

^{n}*→ IR is quasi-convex if and only if the level*

*sets L*

*f*

*(α) := {x ∈ dom f | f (x) ≤ α} are convex for every α ∈ IR.*

* Definition 2.7 Let f*: IR

^{n}*→ IR be a proper function. Then, f is called strictly quasi-convex*

*if for all x, y ∈ dom f with f (x) = f (y), there always holds*

*f(βx + (1 − β)y) < max{ f (x), f (y)} for ∀β ∈ (0, 1).*

By [3, Lemma 3.5.7], if f is lower semicontinuous and strictly quasi-convex, then f is quasi- convex. For a strictly quasi-convex function, we have the following important result, which implies that every local optimal solution of (5) is also a global optimal solution.

**Proposition 2.8 [3, Theorem 3.5.6] Let f**: IR^{n}*→ IR be a proper strictly quasi-convex*
*function. Consider the problem to minimize f(x) subject to x ∈ C, where C is a nonempty*
*convex set in IR*^{n}*. If* *¯x is a local optimal solution, then ¯x is also a global optimal solution.*

* Definition 2.9 Let f*: IR

^{n}*→ IR be a proper function. Then, f is called strongly quasi-convex*

*if for all x, y ∈ dom f with x = y, there always holds*

*f(βx + (1 − β)y) < max{ f (x), f (y)} for ∀β ∈ (0, 1).*

It can be shown that every strongly quasi-convex function is strictly quasi-convex, and every strongly quasi-convex function is quasi-convex even without semicontinuity assumption.

*When f(x) is strongly quasi-convex, the problem (*5) has the unique global optimal solution.

**3 Distance-like function d**_{φ}**and its properties**

In this section, we present the definition of the kernel*φ and investigate the properties of*
*the bivariate function d** _{φ}* induced by

*φ via formula (4). We start with the assumptions on*the function

*ϕ, needed to define the kernel φ. Let ϕ: IR → (−∞, +∞] be a closed proper*convex function with dom

*ϕ = ∅ and domϕ ⊆ [0, +∞). We assume that*

(i) *ϕ is twice continuously differentiable on int(domϕ) = (0, +∞);*

(ii) *ϕ is strictly convex on its domain;*

(iii) lim_{t}_{→0}+*ϕ*^{}*(t) = −∞;*

(iv) *ϕ(1) = ϕ*^{}*(1) = 0 and ϕ*^{}*(1) > 0.*

In the rest of this paper, we denote by* the class of functions satisfying (1)–(4).*

Given*ϕ ∈ , we define the following two subclasses of :*

1=

*ϕ ∈ : ϕ*^{}*(1)(1 − 1/t) ≤ ϕ*^{}*(t) ≤ ϕ*^{}*(1) ln t, ∀t > 0*

(8) and

2=

*ϕ ∈ : ϕ*^{}*(1)(1 − 1/t) ≤ ϕ*^{}*(t) ≤ ϕ*^{}*(1)(t − 1), ∀t > 0*

*.* (9)

*Since ln t≤ t − 1 for any t > 0 and ϕ*^{}*(1) > 0, clearly, *1*⊆ *2*⊆ . The assumptions on*

1and2are very mild. It is not hard to verify that the following functions
*ϕ*1*(t) = t ln t − t + 1, dom ϕ = [0, +∞),*
*ϕ*2*(t) = − ln t + t − 1, dom ϕ = (0, +∞),*

*ϕ*3*(t) = (*√

*t− 1)*^{2}*, dom ϕ = [0, +∞)*

are all in1, and consequently belong to2. The first example *ϕ*1 plays an important
role in the convergence analysis of our first class of algorithms that will be studied in the

next section. As mentioned in the introduction, the*ϕ-divergence for ϕ = ϕ*1 is exactly the
Kullback–Leibler entropy function, given by

*H(x, y) := d**ϕ**(x, y) =*

*n*
*j=1*

*x**j*ln(x*j**/y**j**) + y**j**− x**j**,* (10)

whose domain can be continuously extended to IR_{+}* ^{n}* × IR

_{++}

*by using the convention that 0 ln 0*

^{n}*= 0. The following lemma states some useful properties of H(x, y), and since their*proofs are elementary by use of (10), we here omit them.

**Lemma 3.1 Let H**(·, ·) be defined as in (10). Then, we have the following results.

*(a) The level sets of H(x, ·) are bounded for all x ∈ IR*^{n}_{+}*.*

*(b) If{y** ^{k}*} ⊂ IR

^{n}_{++}

*converges to y*∈ IR

^{n}_{+}

*, then lim*

_{k→+∞}*H(y, y*

^{k}*) = 0.*

*(c) If{z** ^{k}*} ⊂ IR

_{+}

^{n}*, {y*

*} ⊂ IR*

^{k}

^{n}_{++}

*are sequences such that{z*

^{k}*} is bounded, lim*

*k→+∞*

*y*

^{k}*= y*

*and lim*

*k*→+∞

*H(z*

^{k}*, y*

^{k}*) = 0, then lim*

*k*→+∞

*z*

^{k}*= y.*

With the above assumptions on*ϕ, we now give the definition of the kernel φ involved in*
*the function d** _{φ}*. Given

*ϕ ∈ and the parameters µ > 0 and ν ≥ 0, let φ : IR → (−∞, +∞]*

be a closed proper convex function defined by
*φ(t) := µϕ(t) +ν*

2*(t − 1)*^{2}*.* (11)

It is not difficult to verify that*φ satisfies the properties listed in (i)–(iv), and consequently*
*φ ∈ . Particularly, φ will be strongly convex on its domain if ν > 0. This implies that the*
objective function of the subproblem (6), i.e., f*(x) + λ*_{n}

*i=1**(x*_{i}^{k}*)*^{2}*φ(x**i**/x*_{i}^{k}*) will be strictly*
convex on IR^{n}_{++} if the parameter*λ is set to be sufficiently large, although f (x) itself is*
*quasi-convex. That is to say, the proximal term d*_{φ}*(·, ·) plays a convexification role in the*
quasi-convex subproblem (6), and moreover, the convexification role becomes stronger as
the parameter*λ increases. In fact, from the computational results in Sect.*5, we may see that
*the proximal term shows a good convexification role for the quasi-convex function f(x),*
even for a very small*λ.*

*In what follows, we will concentrate on the properties of the bivariate function d** _{φ}*.

**Lemma 3.2 Given a**ϕ ∈ and the parameters µ > 0, ν ≥ 0, and let φ be the kernel*defined by (11) and d*

_{φ}*(·, ·) be the function induced by φ via formula (4). Then,*

*(a) d*_{φ}*is a homogeneous function of order 2, i.e., d*_{φ}*(αx, αy) = α*^{2}*d*_{φ}*(x, y) for ∀α > 0.*

*(b) For a fixed y*∈ IR^{n}_{++}*, the function d*_{φ}*(·, y) is strictly convex over IR*^{n}_{++}*. If, in addition,*
*ν > 0, then d*_{φ}*(·, y) is strongly convex on IR*_{++}^{n}*.*

*(c) For any(x, y) ∈ IR*^{n}_{++}× IR^{n}_{++}*, d*_{φ}*(x, y) ≥ 0, and d*_{φ}*(x, y) = 0 if and only if x = y.*

*(d) For any fixed z* ∈ IR_{++}^{n}*, the level sets L(z, γ ) := {x ∈ IR*_{++}^{n}*: d**φ**(x, z) ≤ γ } are*
*bounded for allγ ≥ 0.*

*(e) Ifϕ ∈ *1*or*2*, and{y** ^{k}*}

*k∈N*⊆ IR

_{++}

^{n}*converges to¯y ∈ IR*

_{+}

^{n}*, then for any fixed x*∈ IR

_{++}

^{n}*,*

*the sequence{d*

*φ*

*(x, y*

^{k}*)}*

*k∈N*

*is bounded.*

*Proof* *The properties in (a) and (b) are clear from the definition of d** _{φ}*given by (4).

(c) Note that*φ(t) is strictly convex and moreover φ*^{}*(1) = µϕ*^{}*(1) = 0 due to (iv). Hence,*
*φ(t) ≥ φ(1) = 0 and φ(t) = 0 iff t = 1.*

*This implies that d*_{φ}*(x, y) ≥ 0 for ∀(x, y) ∈ IR*^{n}_{++}× IR^{n}_{++}*, and d*_{φ}*(x, y) = 0 iff x = y.*

(d) To prove the result, it is enough to consider the one-dimensional case, i.e., to show that
*h*_{ζ}*(t):= ζ*^{2}*φ(t/ζ ) for ζ > 0 has bounded level sets, which in turn is equivalent to showing*
that*φ has bounded level sets. Note that {t : φ(t) ≤ 0} = {1}. Therefore, the conclusion*
follows from [18, Corrollary 8.7.1].

(e) From the definitions of*φ and d** _{φ}*, we have that

*d*_{φ}*(x, y*^{k}*) =*

*n*
*i*=1

⎡

*⎣µ(y*_{i}^{k}*)*^{2}*ϕ*
*x*_{i}

*y*_{i}^{k}

+*ν*
2*(y*_{i}^{k}*)*^{2}

*x*_{i}*y*^{k}* _{i}* − 1

2⎤

⎦

=

*n*
*i=1*

*µ(y*_{i}^{k}*)*^{2}*ϕ(x**i**/y*_{i}^{k}*) +ν*

2*(x**i**− y*_{i}^{k}*)*^{2}
*.*

If*ϕ(t) is bounded above for any t > 0, then the conclusion is obvious. Otherwise, we discuss*
the following two cases:

**Case (1)***¯y**i**> 0 for each i ∈ {1, 2, . . . , n}. Since {y*_{i}* ^{k}*}

*k∈N*

*→ ¯y*

*i*

*for each i , the proof follows*directly from the continuity of

*ϕ.*

* Case (2) there exists an index i*0

*∈ {1, 2, . . . , n} such that ¯y*

*i*0 = 0. By the given assumptions and Case (1), it suffices to prove that the sequence

*{(y*

_{i}

^{k}_{0}

*)*

^{2}

*ϕ(x*

*i*

*/y*

_{i}

^{k}_{0}

*)} is bounded above. For*

*any k∈ N, using the convexity of ϕ and the fact that ϕ(1) = 0, we have that*

0*≥ ϕ(x**i**/y*_{i}^{k}_{0}*) + ϕ*^{}*(x**i**/y*_{i}^{k}_{0}*)*

1*− x**i**/y*^{k}_{i}_{0}
*.*
Multiplying the inequality with*(y*^{k}_{i}_{0}*)*^{2}readily yields that

*(y*_{i}^{k}_{0}*)*^{2}*ϕ(x**i**/y*_{i}^{k}_{0}*) ≤ (y*_{i}^{k}_{0}*)*^{2}*ϕ*^{}*(x**i**/y*_{i}^{k}_{0}*)*

*x**i**/y*_{i}^{k}_{0}− 1

*= (y*_{i}^{k}_{0}*)ϕ*^{}*(x**i**/y*_{i}^{k}_{0}*)*

*x**i**− y*_{i}^{k}_{0}
*,*
which in turn implies that

*(y*_{i}^{k}_{0}*)*^{2}*ϕ(x**i**/y*_{i}^{k}_{0}*) ≤**(y**i*^{k}_{0}*)ϕ*^{}*(x**i**/y*_{i}^{k}_{0}*)(x**i**− y*_{i}^{k}_{0}*)**.*

If*ϕ ∈ *2, then it follows from (9) that

*ϕ*^{}*(1)y*_{i}^{k}_{0}*(1 − y*_{i}^{k}_{0}*/x**i**) ≤ y*_{i}^{k}_{0}*ϕ*^{}*(x**i**/y*_{i}^{k}_{0}*) ≤ ϕ*^{}*(1)(x**i**− y*_{i}^{k}_{0}*).*

Combining the last two inequalities immediately gives that
*(y*_{i}^{k}_{0}*)*^{2}*ϕ(x**i**/y*_{i}^{k}_{0}*) ≤ max*

*ϕ*^{}*(1)*

*x**i**− y*_{i}^{k}_{0}_{2}

*, ϕ*^{}*(1)y*_{i}^{k}

0

*x**i*

*x**i**− y*_{i}^{k}_{0}_{2}
*.*

This together with the given assumptions shows that*{(y*_{i}^{k}_{0}*)*^{2}*ϕ(x**i**/y*_{i}^{k}_{0}*)} is bounded above*
for any*ϕ ∈ *2, and consequently the sequence*{d**φ**(x, y*^{k}*)}**k∈N* is bounded. Noting that

1*⊆ *2, the sequence*{d**φ**(x, y*^{k}*)}**k∈N*is also bounded for*ϕ ∈ *1.
Lemma3.2*(a)–(c) state that d** _{φ}* defined by (4) is a convex second-order homogeneous
distance-like function. Thus, in analogy with the Euclidean distance, we can define the pro-

*jection of a point y, denoted by*

*ˆx(y), to a closed convex set S ⊆ IR*

^{n}*with respect to d*

*, which is characterized as the solution of the following problem*

_{φ}inf

*d*_{φ}*(x, y): x ∈ S*

*.* (12)

The existence of *ˆx(y) is guaranteed by Lemma*3.2(d). For this projection, we have the
following similar results to the Euclidean projection.

**Lemma 3.3 Let S be a closed convex subset of IR**^{n}*and y*∈ IR^{n}*be a point not in S. Then*
*ˆx(y) is the projection of y on S with respect to d**φ**if and only if*

*x− ˆx(y), −∇*1*d*_{φ}*( ˆx(y), y)*

*≤ 0, ∀x ∈ S.* (13)

*Proof* Note that problem (12) is equivalent to inf{d*φ**(x, y) + δ(x | S) : x ∈ IR** ^{n}*}, where

*δ(· | S) denotes the indicator function of the set S. By [19, Theorem 27.4],*

*ˆx(y) solves the*unconstrained optimization problem if and only if the inequality (13) holds. Thus, the proof

is completed.

*Finally, we present a favorable property of d** _{φ}* with

*ϕ ∈*1 or2, which will play a crucial role in the convergence analysis of algorithms to be studied in the next section.

**Lemma 3.4 Given a**ϕ ∈ and the parameters µ > 0, ν ≥ 0, and let φ be the kernel*defined as in (11). Then, for any a, b ∈ IR*^{n}_{++}*and c*∈ IR^{n}_{+}*, we have the following results:*

*(a) Ifν = 0 and ϕ ∈ *1*, thenc − b, ∇*1*d*_{φ}*(b, a) ≤ µϕ*^{}*(1) max*1*≤ j≤n**{a**j**}[H(c, a) −*
*H(c, b)].*

*(b) Ifν ≥ µϕ*^{}*(1) > 0 and ϕ ∈ *2*, thenc − b, ∇*1*d*_{φ}*(b, a) ≤ θ(c − a*^{2}*− c − b*^{2}*)*
*withθ = (ν + µϕ*^{}*(1))/2.*

*Proof* (a) Since*ϕ ∈ *1, we have from (8) that

*ϕ*^{}*(t) ≤ ϕ*^{}*(1) ln t for any t > 0.*

*Setting t= b**j**/a**j*in the inequality, we then obtain that

*c*_{j}*a*_{j}*ϕ*^{}*(b**j**/a**j**) ≤ c**j**a*_{j}*ϕ*^{}*(1) ln(b**j**/a**j**), j = 1, 2, . . . , n.* (14)
On the other hand, it follows from (8) that

*−ϕ*^{}*(t) ≤ −ϕ*^{}*(1)(1 − 1/t), ∀t > 0.*

*Substituting t= b**j**/a**j* *into the inequality and multiplying with a**j*gives

*− b**j**a*_{j}*ϕ*^{}*(b**j**/a**j**) ≤ a**j**ϕ*^{}*(1)(a**j**− b**j**), j = 1, 2, . . . , n.* (15)
Define

*
*^{}*(a, b):= (a*1*ϕ*^{}*(b*1*/a*1*), . . . , a**n**ϕ*^{}*(b**n**/a**n**))*^{T}*, ∀a, b ∈ IR*^{n}_{++}*.*
Then, adding the inequalities (14) and (15) and summing over j *= 1, . . . , n gives*

*c− b,
*^{}*(a, b)*

*≤ ϕ*^{}*(1)*

⎡

⎣^{n}

*j*=1

*a*_{j}

*c** _{j}*ln

*(b*

*j*

*/a*

*j*

*) + a*

*j*

*− b*

*j*

⎤

⎦

*≤ ϕ*^{}*(1) max*

1≤ j≤n*{a**j*}

⎡

⎣^{n}

*j*=1

*c** _{j}*ln

*(b*

*j*

*/a*

*j*

*) + a*

*j*

*− b*

*j*

⎤

⎦

*= ϕ*^{}*(1) max*

1≤ j≤n*{a**j**} [H(c, a) − H(c, b)] .*

Note that∇1*d*_{φ}*(b, a) = µ
*^{}*(a, b), and hence we obtain the result from the last inequality.*

(b) The proof is similar to [2, Lemma 3.4]. For completeness, we here include it. Since
*ϕ ∈ *2, the inequality (15) still holds. On the other hand, we have from (9) that

*ϕ*^{}*(t) ≤ ϕ*^{}*(1)(t − 1), ∀t > 0.*

*Substituting t= b**j**/a**j* into the above inequality leads to

*c**j**a**j**ϕ*^{}*(b**j**/a**j**) ≤ c**j**a**j**ϕ*^{}*(1)(b**j**/a**j**− 1) = ϕ*^{}*(1)c**j**(b**j**− a**j**), j = 1, 2, . . . , n. (16)*
Adding the two inequalities (15) and (16), summing over j *= 1, 2, . . . , n, and using the*
definition of*
*^{}*(a, b), we obtain*

*c − b,
*^{}*(a, b) ≤ ϕ*^{}*(1)*

*n*
*j*=1

*c**j**(b**j**− a**j**) + a**j**(a**j**− b**j**)*

*= ϕ*^{}*(1)c − a, b − a.*

Note that∇1*d*_{φ}*(b, a) = µ
*^{}*(a, b) + ν(b − a). Then, the last inequality implies that*

*c − b, ∇*1*d*_{φ}*(a, b) ≤ µϕ*^{}*(1)c − a, b − a + νc − b, b − a.* (17)
Using the identities

*c − a, b − a = (1/2)(c − a*^{2}*− c − b*^{2}*+ b − a*^{2}*)*
and

*c − b, b − a = (1/2)(c − a*^{2}*− c − b*^{2}*− b − a*^{2}*)*
we then from (17) obtain

*c − b, ∇*1*d*_{φ}*(b, a) ≤ θ(c − a*^{2}*− c − b*^{2}*) −*1

2*(ν − µϕ*^{}*(1))b − a*^{2}

*≤ θ(c − a*^{2}*− c − b*^{2}*),*

where the second inequality is due to*ν ≥ µϕ*^{}*(1). Thus, the proof is completed.*

**4 Interior proximal-like methods**

In this section, we consider two classes of proximal-like algorithms based on the second-order
*homogeneous function d** _{φ}* for the quasi-convex optimization problem (5). The two kinds of
algorithms are described as follows, where the RIPM was first proposed by Auslender et al.

[2] for convex minimization problems subject to non-negative constraints.

* Interior Proximal Method (IPM) Letφ be defined as in (*11) with

*µ > 0, ν = 0 and*

*ϕ ∈*1

*. Generate the sequence{x*

*}*

^{k}*k∈N*

*by the iterative scheme (6).*

**Regularized Interior Proximal Method (RIPM) Let**φ be defined as in (11) withν ≥*µϕ*^{}*(1) > 0 and ϕ ∈ *2*. Generate the sequence{x** ^{k}*}

*k*

*∈N*

*by the iterative scheme (6).*

To establish the convergence of IPM and RIPM, throughout this section, we make the following assumptions for the quasi-convex optimization problem (5):

*(A1) dom f* ∩ IR_{++}* ^{n}* = ∅.

(A2) The optimal set of problem (5), denoted by*X*^{∗}, is nonempty and bounded.

In what follows, we concentrate on the convergence of IPM and RIPM. We first prove that they are well-defined, which is a direct consequence of the following lemma.

**Lemma 4.1 Given**µ > 0, ν ≥ 0 and ϕ ∈ , and let φ and d_{φ}*be defined as in (11) and*
(4), respectively. Then, under assumptions (A1) and (A2), the sequence*{x** ^{k}*}

*k*

*∈N*

*generated*

*by the iterative scheme (6) is well defined.*

*Proof* *The proof proceeds by induction. Clearly, when k*= 0, the conclusion holds due to
(6). Assume that x^{k}*is well defined. Let f*^{∗}be the optimal value of problem (5), then

*f(x) + λ**k**d*_{φ}*(x, x*^{k}*) ≥ f*^{∗}*+ λ**k**d*_{φ}*(x, x*^{k}*) for all x ∈ IR*^{n}_{++}*.* (18)
*Let F**k**(x) := f (x) + λ**k**d*_{φ}*(x, x*^{k}*) and denote its level sets by*

*L**F**k**(γ ) := {x ∈ IR*^{n}_{++}*: F**k**(x) ≤ γ } for all γ ∈ IR.*

Then, the inequality in (18) implies that L_{F}_{k}*(γ ) ⊆ L(x*^{k}*, λ*^{−1}_{k}*(γ − f*^{∗}*)). By Lemma*3.2(c),
*the level sets L(x*^{k}*, λ*^{−1}_{k}*(γ − f*^{∗}*)) are bounded for any γ ≥ f*^{∗}, and consequently, the sets
*L**F*_{k}*(γ ) are bounded for any γ ≥ f*^{∗}. Whereas for any*γ ≤ f*^{∗}*, we have L**F*_{k}*(γ ) ⊆* *X*^{∗},
which are obviously bounded due to assumption (A2). The two sides show that the level sets
*of the function F*_{k}*(x) are bounded. Also, F**k**(x) is lower semicontinuous on dom f . Hence,*
*the level sets of F*_{k}*(x) are compact. Now, using the lower semicontinuity of F**k**(x) and the*
*compactness of its level sets, we have that F*_{k}*(x) has a global minimum which may not be*
*unique due to the nonconvexity of f . In such case, x** ^{k+1}*can be arbitrarily chosen among the

*set of minimizers of F**k**(x).*

Next, we investigate the properties of the sequence*{x** ^{k}*}

*k∈N*generated by IPM and RIPM.

To this end, we define the following set
*U* :=

*x*∈ IR_{+}^{n}*| f (x) ≤ inf*

*k∈N* *f(x*^{k}*)*
*.*

From assumptions (A1)–(A2) and Proposition2.6, U is a nonempty closed convex set.

**Lemma 4.2 Let**{λ*k*}*k**∈N**be an arbitrary sequence of positive numbers and{x** ^{k}*}

*k*

*∈N*

*be the*

*sequence generated by IPM. Then, under assumptions (A1)–(A2),*

(a) *{ f (x*^{k}*)}**k∈N**is a decreasing and convergent sequence.*

(b) *{x** ^{k}*}

*k*

*∈N*

*is Fejér convergent to the set U with respect to H .*

*(c) For all x∈ U, the sequence {H(x, x*

^{k}*)}*

_{k∈N}*is convergent.*

*Proof* (a) From Eq.6, x* ^{k+1}*is a global optimal solution of the following problem:

min*x≥0*

*f(x) + λ**k**d*_{φ}*(x, x*^{k}*)*
*and consequently, for any x*∈ IR^{n}_{+}, it follows that

*f(x*^{k+1}*) + λ**k**d*_{φ}*(x*^{k+1}*, x*^{k}*) ≤ f (x) + λ**k**d*_{φ}*(x, x*^{k}*).* (19)
*Setting x= x** ^{k}*in (19), we then obtain that

*f(x*^{k+1}*) + λ**k**d*_{φ}*(x*^{k+1}*, x*^{k}*) ≤ f (x*^{k}*) + λ**k**d*_{φ}*(x*^{k}*, x*^{k}*) = f (x*^{k}*),*
which means that

0*≤ λ**k**d*_{φ}*(x*^{k+1}*, x*^{k}*) ≤ f (x*^{k}*) − f (x*^{k+1}*).*

Hence,*{ f (x*^{k}*)}**k**∈N*is decreasing, and furthermore, convergent due to assumption (A2).

(b) From inequality (19), it follows that for any x*∈ U,*
*d*_{φ}*(x*^{k+1}*, x*^{k}*) ≤ d*_{φ}*(x, x*^{k}*).*

*This implies that x*^{k}^{+1}*is the unique projection of x*^{k}*on U with respect to d** _{φ}*. Therefore, by
Lemma3.3, we have that

*x− x*^{k}^{+1}*, −∇*1*d*_{φ}*(x*^{k}^{+1}*, x*^{k}*)*

*≤ 0, ∀x ∈ U.* (20)

On the other hand, applying Lemma3.4*(a) at the points c= x, a = x*^{k}*, and b= x** ^{k+1}*, we
then obtain that

*H(x, x*^{k}*) − H(x, x*^{k+1}*) ≥* *x − x*^{k+1}*, ∇*1*d*_{φ}*(x*^{k+1}*, x*^{k}*)*

*µϕ*^{}*(1) max*1*≤ j ≤ n**{x*^{k}* _{j}*}

*.*(21) Since

*µϕ*

^{}

*(1) max*1

*≤ j ≤ n*

*{x*

^{k}

_{j}*} > 0, using the inequalities (*20) and (21) yields that

*H(x, x*^{k}*) ≥ H(x, x*^{k+1}*), ∀x ∈ U.*

From Definition2.1, it follows that*{x** ^{k}*}

*k∈N*

*is Fejér convergent to U with respect to H .*

*(c) The proof follows from part (b) and the non-negativity of H .*

**Lemma 4.3 Let**{λ*k*}

*k*

*∈N*

*be an arbitrary sequence of positive numbers and{x*

*}*

^{k}*k*

*∈N*

*be the*

*sequence generated by RIPM. Then, under assumptions (A1) and (A2),*

(a) *{ f (x*^{k}*)}**k**∈N**is a decreasing and convergent sequence.*

(b) *{x** ^{k}*}

*k*

*∈N*

*is Fejér convergent to the set U .*

*(c) For all x∈ U, the sequence {x − x** ^{k}*}

*k∈N*

*is convergent.*

*Proof*

(a) The proof is similar to that of Lemma4.2(a), and we here omit it.

(b) By a similar argument to Lemma4.2(b), we can obtain the inequality (20). On the other
hand, applying Lemma3.4*(b) at the points c= x, a = x*^{k}*, and b= x** ^{k+1}*gives

*x − x*^{k}*, ∇*1*d*_{φ}*(x*^{k}*, x*^{k+1}*) ≤ θ(x − x*^{k}^{2}*− x − x*^{k+1}^{2}*),* (22)
where*θ = (ν + µϕ*^{}*(1))/2. Since θ > 0, using the inequalities (20) and (22) yields that*

*x − x*^{k}^{+1}^{2}*≤ x − x*^{k}^{2}*, ∀x ∈ U.* (23)
By Definition2.1, we thus prove that*{x** ^{k}*}

*k*

*∈N*

*is Fejér convergent to the set U .*

(c) The proof follows from part (b) and the non-negativity of*x − x** ^{k}*.
To now, we have proved that the sequence

*{x*

*}*

^{k}*k*

*∈N*generated by IPM or RIPM is well- defined and satisfies some favorable properties. With these properties, we next establish the convergence results of the proposed algorithms.

**Proposition 4.4 Suppose that assumptions (A1) and (A2) are satisfied. Let**{λ*k*}*k**∈N* *be an*
*arbitrary sequence of positive numbers and{x** ^{k}*}

*k*

*∈N*

*be the sequence generated by IPM.*

*Then, the sequence{x** ^{k}*}

*k∈N*

*converges, and furthermore,*

*(a) if there existλ and ¯λ such that 0 < λ < λ**k**≤ ¯λ for any k, then*
lim inf

*k→+∞**g*_{i}^{k}*≥ 0,* lim

*k→+∞**g*^{k}_{i}*x*_{i}^{k}*= 0, ∀i = 1, 2, . . . , n,* (24)
*where g*^{k}*∈ ˆ∂ f (x*^{k}*) and g*^{k}_{i}*is the i th component of g*^{k}*.*

*(b) If lim*_{k→+∞}*λ**k**= 0, then {x** ^{k}*}

_{k∈N}*converges to a solution of the problem (5).*

*Proof* We first prove that the sequence*{x** ^{k}*}

*k∈N*converges. By Lemma4.2(b),

*{x*

*}*

^{k}*k∈N*is

*Fejér convergent to the set U with respect to H , which implies that*

*{x** ^{k}*}

*k*

*∈N*⊆

*y*∈ IR_{++}^{n}*| H(x, y) ≤ H(x, x*^{0}*)*

for*∀x ∈ U.*

As a consequence,*{x** ^{k}*}

*k∈N*is bounded by Lemma 3.1(a). Thus, there exist an

*¯x and a*subsequence

*{x*

^{k}

^{j}*} of {x*

*}*

^{k}*k*

*∈N*converging to

*¯x. From the lower semicontinuity of f ,*

*j→+∞*lim *f(x*^{k}^{j}*) ≥ f ( ¯x),*
which, together with Lemma4.2(a), implies that

*f( ¯x) ≤ f (x*^{k}*), ∀k ∈ N.*

This shows that*¯x ∈ U. By Lemma*4.2(c), the sequence*{H( ¯x, x*^{k}*)}**k∈N*is then convergent.

In addition, from Lemma3.1(b), we have lim_{k→+∞}*H( ¯x, x*^{k}^{j}*) = 0. From all the above, we*
conclude that*{H( ¯x, x*^{k}*)}**k∈N*is a convergent sequence with a subsequence converging to 0,
and consequently it must converge to 0 itself, i.e., lim*k*→+∞*H( ¯x, x*^{k}*) = 0. Using Lemma*
3.1*(c) with z*^{k}*= x*^{k}*and y*^{k}*= ¯x, we thus prove that {x** ^{k}*}

*k*

*∈N*converges to

*¯x.*

(a) From the iterative formula (6) and Lemma2.3(d), we have that
0*∈ ˆ∂*

*f(x) + λ**k**d*_{φ}*(x, x*^{k}*)*
*(x*^{k+1}*).*

Therefore, by Lemma2.3*(c), there exists g*^{k}^{+1}*∈ ˆ∂ f (x*^{k}^{+1}*) such that*
*λ**k*∇1*d*_{φ}*(x*^{k}^{+1}*, x*^{k}*) = −g*^{k}^{+1}*,*

i.e.,

*µλ**k**x*_{i}^{k}*ϕ*^{}
*x*_{i}^{k}^{+1}

*x*_{i}^{k}

*= −g*^{k+1}_{i}*, i = 1, 2, . . . , n.* (25)
Define the index sets

*I( ¯x) :=*

*i∈ {1, 2, . . . , n} | ¯x**i* *> 0*

*and J( ¯x) :=*

*i∈ {1, 2, . . . , n} | ¯x**i* = 0
*.*
*We next argue the conclusion by the two cases i∈ I ( ¯x) and i ∈ J( ¯x).*

**Case (1) i**∈ I ( ¯x). In this case, lim_{k→+∞}*x*_{i}^{k}^{+1}*/x*_{i}^{k}*= 1 since {x** ^{k}*}

*converges to*

_{k∈N}*¯x. Using*the continuity of

*ϕ*

^{}and

*ϕ*

^{}

*(1) = 0 and recalling that 0 < λ ≤ λ*

*k*

*≤ ¯λ for all k, we then*obtain from (25) that

*k*→+∞lim *g*_{i}^{k}^{+1}*= 0, ∀i ∈ I ( ¯x).* (26)
**Case (2) i**∈ J( ¯x). For every i ∈ J( ¯x), we define the following two index sets:

*J*_{+}* ^{i}* =

*k: x*_{i}^{k+1}*/x*_{i}^{k}*> 1*

and *J*_{−}* ^{i}* =

*k: x*_{i}^{k+1}*/x*_{i}* ^{k}*≤ 1

*.*

Since*ϕ*^{}*(1) = 0 and ϕ*^{}is monotone increasing on its domain, we have from (25) that
*g*_{i}^{k+1}*≤ 0 for ∀k ∈ J*_{+}^{i}*, ∀i ∈ J( ¯x).*

On the other hand, using (25) and the fact that*ϕ ∈ *1*⊆ *2yields that
*g*_{i}^{k+1}*≥ −µϕ*^{}*(1)λ**k**x*_{i}^{k}

*x*_{i}^{k}^{+1}
*x*_{i}* ^{k}* − 1

*≥ −µϕ*^{}*(1)¯λ(x*_{i}^{k+1}*− x*_{i}^{k}*), ∀k ∈ J*_{+}^{i}*.*

Noting that lim_{k→+∞}*(x*_{i}^{k}^{+1}*− x*_{i}^{k}*) = 0, the last two equations imply that*

*k→+∞, k∈J*lim _{+}^{i}*g*_{i}^{k+1}*= 0, ∀i ∈ J( ¯x).* (27)

Furthermore, since*ϕ*^{}*(t) ≤ 0 for any 0 < t ≤ 1 by (ii) and (iv), we have from (*25) that
*g*_{i}^{k+1}*≥ 0, ∀k ∈ J*_{−}^{i}*, ∀i ∈ J( ¯x).* (28)
The inequalities (26)–(28) immediately imply the first part of (24), i.e.,

lim inf

*k*→+∞*g*_{i}^{k}*≥ 0, ∀i = 1, 2, . . . , n.*

Next, let us prove the second part of (24). Using (26) and (27) and the fact that*{x** ^{k}*}

*k∈N*

converges to*¯x, we have only to prove that*

*k→+∞, k∈J*lim _{−}^{i}*g*^{k}_{i}^{+1}*x*_{i}^{k}^{+1}*= 0, ∀i ∈ J( ¯x).*

Considering that

*k→+∞, k∈J*lim _{−}^{i}*x*_{i}^{k}^{+1}*= 0, ∀i ∈ J( ¯x)*

and using the first part of (24), we then have only to prove that the subsequence*{g*^{k}* _{i}*}

_{k∈J}_{−}

*for*

^{i}*each i∈ J( ¯x) is bounded above. Take*0

*> 0 and x ∈ IR*

^{n}_{++}

*∩ dom f with x*

*i*

*>*0for any

*i . Then for k∈ J*

_{−}

*sufficiently large, we have*

^{i}*x**i**− x*_{i}* ^{k+1}*≥0

2*, ∀i ∈ J( ¯x).* (29)

From Definition2.2, we have
*f(x) ≥ f (x*^{k}^{+1}*) +*

*n*
*i*=1

*g*_{i}^{k+1}*(x − x*_{i}^{k+1}*) + o(x − x*^{k}^{+1}*),* (30)

which implies that the subsequence*{g*_{i}* ^{k}*}

_{k}

_{∈J}_{−}

^{i}*is bounded above for i*

*∈ J( ¯x). Indeed, sup-*

*pose the contrary. Then there would exist an i*

_{0}

*∈ J( ¯x) and a subsequence {g*

_{i}

^{k}_{0}

*}*

^{l}

_{k}

_{l}

_{∈J}_{−}

*(with lim*

^{i}*l*→+∞

*k*

*l*

*= +∞) such that*

*l→+∞*lim *g*_{i}^{k}^{l}^{+1}

0 *= +∞, g*_{i}^{k}_{0}^{l}^{+1}*≥ 0.*

Since the sequence*{x** ^{k}*}

*k∈N*is convergent, using the Eq.27–29gives that there exists

*η ∈ IR*

*such that for sufficiently large l,*

*i=i*0

*g*_{i}^{k}^{l}^{+1}*(x**i**− x*_{i}^{k}^{l}^{+1}*) + o(x − x*^{k+1}*) ≥ η.*

Then, from (29) and (30), we obtain

*f(x) ≥ f (x*^{k}^{l}^{+1}*) +*0

2*g*_{i}^{k}^{l}^{+1}

0 *+ η.*

Since lim*l*→+∞ *f(x*^{k}^{l}^{+1}*) ≥ f ( ¯x) and lim**l*→+∞*g*^{k}_{i}^{l}^{+1}

0 = +∞, passing to the limit in the above inequality leads to a contradiction.

(b) From the inequality in (19) and the non-negativity of d* _{φ}*, it follows that

*f(x*

^{k+1}*) ≤ f (x) + λ*

*k*

*d*

_{φ}*(x, x*

^{k}*), ∀x ∈ IR*

^{n}_{++}

*.*

*Taking the limit k*→ +∞ into the inequality and using lim*k→+∞**λ**k* = 0, Lemma3.2(e)
*and the lower semicontinuity of f , we then obtain that*

*f( ¯x) ≤ f (x), ∀x ∈ IR*^{n}_{++}*,* (31)

where*¯x is such that lim**k→+∞**x*^{k}*= ¯x. This implies that ¯x ∈X*^{∗}.
**Proposition 4.5 Suppose that assumptions (A1) and (A2) are satisfied. Let**{λ*k*}*k∈N* *be an*
*arbitrary sequence of positive numbers and{x** ^{k}*}

*k∈N*

*be the sequence generated by RIPM.*

*Then, the sequence{x** ^{k}*}

*k∈N*

*converges, and furthermore,*

*(a) If there existλ and ¯λ such that 0 < λ < λ**k**≤ ¯λ for any k, we have*
lim inf

*k→+∞**g*_{i}^{k}*≥ 0,* lim

*k→+∞**g*^{k}_{i}*x*_{i}^{k}*= 0, ∀i = 1, 2, . . . , n,* (32)
*where g*^{k}_{i}*is same as Proposition*4.4.

*(b) If lim*_{k→+∞}*λ**k**= 0, then the sequence {x** ^{k}*}

*k∈N*

*converges to a solution of problem (5).*

*Proof* First, we prove that the sequence*{x** ^{k}*}

*converges. By Lemma4.3(b),*

_{k∈N}*{x*

*}*

^{k}

_{k∈N}*is Fejér convergent to the set U , which implies that*

*{x** ^{k}*}

*k*

*∈N*⊆

*y*∈ IR^{n}*| x − y ≤ x − x*^{0}

for*∀x ∈ U.*

*Note that the latter set is bounded for any given x*∈ IR* ^{n}*, and therefore, the sequence

*{x*

*}*

^{k}*k∈N*

is bounded and there exist an*ˆx and a subsequence {x*^{k}^{j}*} of {x** ^{k}*}

*k∈N*converging to

*ˆx. Using*a similar argument to the first part of Proposition4.4, we can prove that

*ˆx ∈ U. Thus, by*Lemma4.3(c), the sequence

*{x*

^{k}*− ˆx}*

*k*

*∈N*is convergent. Since

*{x*

^{k}*} ∈ IR*

^{j}

^{n}_{++}converges to

*ˆx ∈ IR*

_{+}

*, we have that*

^{n}*ˆx − x*

^{k}

^{j}*→ 0, and consequently ˆx − x*

*→ 0, which implies*

^{k}*that the limit point is unique and x*

^{k}*→ ˆx.*

(a) From the iterative formula (6) and Lemma2.3(d), we have
0*∈ ˆ∂*

*f(x) + λ**k**d*_{φ}*(x, x*^{k}*)*
*(x*^{k+1}*).*

*Therefore, there exists g*^{k}^{+1}*∈ ˆ∂ f (x*^{k}^{+1}*) such that*

*λ**k*∇1*d*_{φ}*(x*^{k}^{+1}*, x*^{k}*) = −g*^{k}^{+1}*,*
*i.e., for each i= 1, 2, . . . , n,*

*g*^{k}_{i}^{+1}*= −µλ**k**x*_{i}^{k}*ϕ*^{}*(x*_{i}^{k}^{+1}*/x*_{i}^{k}*) − νλ**k**(x*_{i}^{k}^{+1}*− x*^{k}_{i}*).* (33)
Since*ϕ ∈ *2, we have from (9) that for each i*= 1, 2, . . . , n,*

*−µλ**k**x*_{i}^{k}*ϕ*^{}

*x*_{i}^{k+1}*x*_{i}^{k}

*≥ −µλ**k**ϕ*^{}*(1)x*_{i}^{k}

*x*_{i}^{k+1}*x*_{i}* ^{k}* − 1

*≥ −µλ**k**ϕ*^{}*(1)(x*_{i}^{k+1}*− x*_{i}^{k}*).*

Combining the last two inequalities then yields that

*g*_{i}^{k}^{+1}*≥ λ**k**(µϕ*^{}*(1) + ν)(x*_{i}^{k}*− x*_{i}^{k}^{+1}*) = 2θλ**k**(x*_{i}^{k}*− x*_{i}^{k}^{+1}*), i = 1, 2, . . . , n,* (34)