A UV-Method for a Class of Constrained Minimized Problems of Maximum Eigenvalue Functions

In this paper, we apply the UV-algorithm to solve the constrained minimization problem of a maximum eigenvalue function which is the composite function of an affinematrix-valuedmapping and its maximum eigenvalue. Here, we convert the constrained problem into its equivalent unconstrained problem by the exact penalty function. However, the equivalent problem involves the sum of two nonsmooth functions, which makes it difficult to applyUV-algorithm to get the solution of the problem. Hence, our strategy first applies the smooth convex approximation of maximum eigenvalue function to get the approximate problem of the equivalent problem. Then the approximate problem, the space decomposition, and the U-Lagrangian of the object function at a given point will be addressed particularly. Finally, theUV-algorithmwill be presented to get the approximate solution of the primal problem by solving the approximate problem.


Introduction
The eigenvalue optimization problems have attracted wide attention to the nonsmooth optimization.Such problems arise from many applications such as signal recovery [1], shape optimization [2], and robotics [3].Therefore, the research on methods for solving such problems plays an important role in enriching the blend of classical mathematical techniques and contemporary optimization theory.Various methods have been proposed to deal with such problems; for example, the bundle method was used by Helmberg and Oustry to solve a class of unconstrained maximum eigenvalue optimization problems [4].Recently, Oustry applied U-Newton algorithm to solve the maximum eigenvalue optimization problem [5].However, this method must satisfy the transversality condition.In this paper, we design a UV-algorithm which does not satisfy the strict condition above to solve the constrained maximum eigenvalue optimization problem approximately.Here, we focus our attention on the following mode problem particularly: min  1 ( ()) s.t.  () ≤ 0,  = 1, 2, . . ., ,

(𝑃)
where  1 (()) is the maximum eigenvalue function and the mapping () fl  0 + A() is affine,  0 ∈   is given, A :   →   is a linear operator, and   is the space of × symmetric matrices.Consider an exact penalty function associated with () as follows:  () fl  1 ( ()) +  max { 0 () ,  1 () , . . .,   ()} , where  0 () ≡ 0, ∇ 0 () ≡ 0, and  > 0 is a penalty parameter.For  large enough, it is well known that the problem () is equivalent to the following form: It is known that the UV-decomposition theory must be applied on the condition that the dimension of the V-space is not full dimensional.Since () inherits the nondifferentiability of  1 (()) and the function max{  }, it is difficult to apply UV-decomposition theory to ( 1 ) in that the Vspace of () at a given point is full dimensional.Hence, it is imperative to consider the smooth approximation function   () [6] to the function  1 (()).Then the approximate problem of ( 1 ) is given as follows: where   () fl   () +  max{ 0 (),  1 (), . . .,   ()}.Thus the problem ( 2 ) can be solved by UV-algorithm and we can get the approximate solution of the problem () at the same time.
The rest of the paper is organized as follows.Section 2 introduces three equivalent UV-space decomposition definitions of   (), associated with a given  ∈   .The U-Lagrangian of   () and relevant property will be addressed more detailedly in Section 3. Section 4 is devoted to the UV-algorithm for solving the approximate problem and the convergence analysis of the method.Finally, Section 5 gives some conclusive comments.
To be convenient for explanation, we give the set of the act indicators throughout the paper and set The solution of the problem () depends on the study of the objective function of problem ( 2 ).The UV-space decomposition theory of   () will be shown firstly.

UV-Space Decomposition for 𝑃 𝜀 (𝑥)
Firstly, we can easily obtain the description of the subdifferential about   () as follows: and the relative interior of   () We start by defining a decomposition of space   = U ⊕ V, associated with a given  ∈   .We give three definitions for the subspaces U and V as follows.
Definition 1. (i) Define U 1 as the subspace where    (, ⋅) is linear and take V 1 fl U ⊥ 1 , and since    (, ⋅) is sublinear, we have (ii) Define V 2 as the subspace parallel to the affine hull of   (); in other words, where  ∈   () is arbitrary and take U 2 fl V ⊥ 2 .
(iii) Define U 3 and V 3 as the normal and tangent cones to   () at a given point  ∘ ; that is, In the meantime, U 3 and V 3 are subspaces.
Theorem 2. In Definition 1, we have the following: (i) The subspace U 3 is actually given by and is independent of the particular  ∘ ∈ ().
Next, we only need to establish the converse case.Let  ∈  () ( ∘ − ∇  ()) and   ∈ () and it suffices to prove By the definition of relative interior, there exists a positive constant  such that Then the result (i) is proved.
The solution of problem () is not only based on the UVspace decomposition of   () but also based on the study of the U-Lagrangian of   (), which will be shown next.

The U-Lagrangian of 𝑃 𝜀 (𝑥)
Let ∇  () fl g = gU ⊕ gV , let  2 fl ∇ 2   () be a positive semidefinite matrix, and let Û be a basis matrix for U. ∀  ∈ (), we define the U-Lagrange function of   () as follows: Associated with (19) we have the set of minimizers In the following paragraphs, a series of theorems and corollaries will be given to specify the property of  U,  and the expansions of   ().

Theorem 3. By the definition of 𝐿 U,𝑃 𝜀 , we have the following conclusions:
(i)  U,  (, ) is a proper convex function.
Theorem 4. Let  satisfy () ̸ = 0.Then, ∀ ∈ (), the subdifferential of  U,  at this  has the expression In particular,  U,  is differentiable at 0, with ∇ U,  (0, ) =   U + gU .V , V⟩ V } has a generalized Hessian  1 at  = 0 and   ∈ ().For  ∈ U and  ∈  ⊕ (), it holds that The proofs of the above theorems and corollary are similar to [7] and here we ignore the details of them.
Based on the study of UV-space decomposition theory and the U-Lagrangian of   (), the UV-algorithm which can solve the problem ( 2 ) will be addressed in the next section.

The UV-Method
Depending on the UV-theory mentioned above, the constrained minimization problem of maximum eigenvalue function has been converted into the convex minimization problem which can be solved by the UV-algorithm in [8].Hence, we apply the UV-algorithm in [8] and do some appropriate modifications for solving the problem ( 2 ).
In this section, some definitions and two quadratic programming problems will be denoted for easy understanding.
Given a tolerance  ∈ (0, 1/2], a prox-parameter  > 0, and a prox-center  ∈   , to find -approximation of   (), our bundle subroutine accumulates information from the candidates {  } ∈B , where B fl { :   = }.Definition 8. Let ,   ∈   ,  ∈ B,   ∈   (), and the linearization error is defined by Definition 9. Given a positive scalar parameter , the proximal point function depending on   () is defined by for  ∈   . (25) The first quadratic programming subproblem ( − ) has the following form and properties; see [9].The problem  (29) The second quadratic programming subproblem is Similar to (28), the respective solutions, denoted by (, ) and , satisfy Since the need of the algorithm, the solution of the problem ( − ) will be applied to get the matrix Ũ.Firstly, define an active index by Bact fl { ∈ B :  =   ( − )}.Then, from (32),  = − ⊤  s for all  ∈ Bact , so for all such  and for a fixed  ∈ Bact .Define a full-column rank matrix Ṽ by choosing the largest number of indices  satisfying (33) such that the corresponding vectors   −   are linearly independent and by letting these vectors be the columns of Ṽ.Then let Ũ be a matrix where columns form an orthogonal basis for the null-space of Ṽ⊤ with Ũ =  if Ṽ is vacuous.
For convenience, in the sequel we denote the output from these calculations by The algorithm depending on the above quadratic programming problems is given as follows.
Step 0. Choose positive parameters , , and  with  < 1.Let  0 ∈   and  0 ∈   ( 0 ), respectively, be an initial point and subgradient.Also, let  0 be a matrix with orthogonal dimensional columns estimating an optimal U-basis.Set  0 =  0 and  fl 0.
Step 2. Choose an   ×  positive definite matrix   , where   is the number of columns of   and   approximating a basis for V(()) ⊤ .For  which is a minimizer of   (), () =  +  ⊕ V(), where V fl  dim U  →  dim V is a  2 -function satisfying V() ∈ () for all  ∈ ri  (). is a basis matrix of V and   is the approximation of ∇ 2  U,  (, 0).
Step 6. Replace  by  + 1 and go to stopping test.
Next, we will show the convergence of Algorithm 10.From here on, we assume that  = 0 and that Algorithm 10 does not terminate.When the primal track at the initial point exists, firstly, it shows that if some execution of the bundle procedure in Algorithm 10 continues indefinitely, there is convergence to a minimizer of   .
Theorem 11.If the bundle procedure does not terminate, that is, if (37) never holds, then the sequence of p-values converges to   () and   () minimizes   ().If the procedure terminates with s = 0, the corresponding p equals   () and minimizes   ().In both of these cases   () −  ∈ V(  ()).
Next theorem shows minimizing convergence from any initial point without assuming the existence of a primal track.
Here we assume that all executions of bundle procedure terminate.
(ii) Now suppose   is bounded below and  is any accumulation point of {  }.Then, because ‖  ‖ and {  } converge to 0 by item (i), (42) together with the continuity of   implies that   () ≤   () for all  ∈   and (ii) is proved.
In order to obtain convergence of the whole sequence   , we need the concept of a strong minimizer.
Definition 13.We say that  is a strong minimizer of   if 0 ∈ ri  () and the corresponding U-Lagrangian  U,  (, 0) has a Hessian at  = 0 that is positive definite.