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let @xmath0 be a finite - dimensional linear vector space with the dual space @xmath1 as the space of all linear function on @xmath0 .
we assume @xmath2 is a proper , @xmath3-strongly convex ( @xmath4 for strongly convex case and @xmath5 for convex case ) , and lower semicontinuous function satisfying f(x)-f(y ) _ *
l _ x - y^   x , y v , where @xmath6 denotes the gradient of @xmath7 at @xmath8 for @xmath9 $ ] or any subgradient of @xmath7 at @xmath8 ( @xmath10 ) for @xmath11 .
let the function @xmath12 be simple , proper , @xmath13-strongly convex ( @xmath14 ) , and lower semicontinuous function .
we consider the structured convex minimization problem    ll &   h(x):=f(x)+(x ) + &   x c ,    where @xmath15 is a simple , nonempty , closed , and convex set . by ( [ e.holder ] ) , we have @xmath16 , i.e. , @xmath7 can be smooth with lipschitz continuous gradients ( @xmath17 ) , weakly smooth problems with hlder continuous gradients ( @xmath180,1[}$ ] ) , or nonsmooth with bounded variation of subgradients ( @xmath19 ) .
hence the objective @xmath20 is @xmath21-strongly convex with @xmath22 .
we assume that the first - order black - box oracle of the objective @xmath20 is available .      over the past few decades , due to the dramatic increase in the size of data for many applications , first - order methods
have been received much attention thanks to their simple structures and low memory requirements .
the efficiency of first - order methods can be poor ( a large number of function values and subgradients is needed ) for solving the general convex problems if the structure of the problem is not available . as a result , to develop practically appealing schemes ,
it is necessary to make an additional restriction on problem classes . in particular
, developing efficient methods for solving large - scale convex optimization problems is possible if the underlying objective has a suitable structure and the domain is simple enough .
convexity and level of smoothness are two important factors playing key roles in construction of efficient schemes for such structured optimization problems .
let @xmath23 be an optimizer of ( [ e.gfun ] ) and @xmath24 be an approximate solution given by a first - order method .
we call @xmath24 an @xmath25-solution of ( [ e.gfun ] ) if @xmath26 , for a prescribed accuracy parameter @xmath27 . in 1983 , nemirovski & yudin in @xcite derived optimal worst - case complexities for first - order methods to achieve an @xmath25-solution for several classes of convex problems ( see table [ t.complexity ] ) .
if a first - order scheme attains the worst - case complexity of a class of problems , it is called optimal .
a special feature of these methods is that the corresponding complexity does not depend explicitly on the problem dimension . from practical point of view , studying the effect of an uniform boundedness of the complexity is very attractive and such methods are highly recommended when the prescribed accuracy @xmath25 is not too small , whereas the dimension of problem is considerably large .
[ t.complexity ]    .list of the best known complexities of first - order methods for several classes of problems with respect to levels of smoothness and convexity ( cf .
@xcite ) [ cols="<,^,^ " , ]
in this paper , we propose several novel ( sub)gradient methods for solving large - scale convex composite minimization .
more precisely , we give two estimation sequences approximating the objective function with some local and global information of the objective . for each of the estimation sequences , we give two iterative schemes attaining the optimal complexities for smooth , nonsmooth , weakly smooth , and smooth strongly convex problems .
these schemes are optimal up to a logarithmic factors for nonsmooth strongly convex problems , and for weakly smooth strongly convex problems they attain a much better complexity than the complexity for weakly smooth convex problems . for each estimation sequence ,
the first scheme needs to know about the level of smoothness and the hlder constant , while the second one is parameter - free ( except for the strong convexity parameter which we set zero if it is not available ) at the price of applying a backtracking line search .
we then consider solutions of the auxiliary problems appearing in these four schemes and study the important cases appearing in applications that can be solved efficiently either in a closed form or by a simple iterative scheme . considering some applicationsin the fields of sparse optimization and machine learning , we report numerical results showing the encouraging behavior of the proposed schemes . + * acknowledgement .
* i would like to thank arnold neumaier for his useful comments on this paper .
golub , t. , slonim , d. , tamayo , p. , huard , c. , gaasenbeek , m. , mesirov , j. , coller , h. , loh , m. , downing , j. , caligiuri , m. : molecular classification of cancer : class discovery and class prediction by gene expression monitoring , science 286 , 531536 ( 1999 )                            nesterov , y. : a method of solving a convex programming problem with convergence rate @xmath29 , doklady an sssr ( in russian ) , 269 ( 1983 ) , 543547 .
english translation : soviet math . dokl .
, * 27 * , 372376 ( 1983 )
this paper discusses several ( sub)gradient methods attaining the optimal complexity for smooth problems with lipschitz continuous gradients , nonsmooth problems with bounded variation of subgradients , weakly smooth problems with hlder continuous gradients . 
 the proposed schemes are optimal for smooth strongly convex problems with lipschitz continuous gradients and optimal up to a logarithmic factor for nonsmooth problems with bounded variation of subgradients . more specifically , we propose two estimation sequences of the objective and give two iterative schemes for each of them . in both cases , 
 the first scheme requires the smoothness parameter and the hlder constant , while the second scheme is parameter - free ( except for the strong convexity parameter which we set zero if it is not available ) at the price of applying a nonmonotone backtracking line search . a complexity analysis for all the proposed schemes 
 is given . 
 numerical results for some applications in sparse optimization and machine learning are reported , which confirm the theoretical foundations .    
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