A Projection and Contraction Method for P-Order Cone Constraint Stochastic Variational Inequality Problem ()
1. Introduction
Variational inequality has important applications in many aspects such as physics, economic equilibrium theory, cybernetics, engineering, optimization, etc. [1] [2] [3]. Since there are many random factors that cannot be ignored in real life, such as weather, demand, price, etc., stochastic variational inequality has become a research hot-spot of many scholars in recent decades. In this paper, we consider the stochastic variational inequality problem with p-order cone (abbrevd.POSVI), which is to find
, such that
(1)
where
denotes the Euclidean inner product,
is a random variable defined in probability
,
is the expectation of
,
is a given mapping,
,
is a p-order cone,
is the dual cone of
, which can be expressed as [4]:
where
and satisfies
.
and
are both closed convex cone,
is not a self-dual cone when
, in other words, the
is not a symmetric cone for
. It is clear that the p-order cone is the second-order cone with n-dimension when
, which means:
Thus, the POSVI in this paper can be regarded as an extension of the second-order constrained stochastic variational inequality problem. The research of theory and algorithm has made great progress in recent decades. Our research focuses on transforming solving the POSVI into finding the zero point of equation and constructing optimization algorithm under certain condition. Hence, the work below is down first.
Assume that a closed convex subset of
is C and
, then the projection of x on C is [5]:
(2)
Refer to reference [6],
is the solution of problem (1) if and only if for any
, the equation holds below
(3)
Define the residual of the equation as
(4)
then, solving POSVI only needs to find a zero point of
.
In the following, we concentrate on constructing algorithm to solve the problem. There are some optimization algorithms that have been applied to stochastic optimization problem, for example, Korpelevich and Antipin study the outer gradient projection algorithm and every iteration should compute the projection twice, see [7] [8]. On the basis of previous research, Tseng proposes a gradient projection algorithm and every iteration only should compute a projection one time, see [9]. Different from the first two algorithms, Duong and Yekini receive step size through Amjo-type search method, which can decrease the difficulty of [9] to estimate the Lipschitz constant, see [10] [11]. Under a mapping is monotone and Lipschitz continuous, Yang and Liu give a projection algorithm and the better of which is that the compute of the step size does not depend on Amjo-like search, see [12] [13] [14]. Combined the above algorithms, we propose a projection and contraction method.
The organizational framework of this paper is as follows: In part 2, we introduce some definitions and conclusions about the p-order cone. In part 3, the basic idea of sample average approximation method is shown and the revelent conclusions are given. In part 4, a projection and contraction method is proposed and it is proved that the sequence generated by the method converges to the real solution. In part 5, a numerical example is given, and the method is applied to solve it. The numerical results claim that the method is effective. In part 6, the work of this paper is summarized and future research topic is given.
2. Preliminaries
In this section, we give some basic concepts and conclusions related to p-order cone in order to facilitate the following research.
Definition 2.1 ( [15] ). Let
be Lipschitz continuous then for any
, there exist a constant
, such that
Definition 2.2 ( [15] ). Let
. Then for any
, the mapping h is said to be
(a) monotoneif
(b) strictlymonotone if
(c) stronglymonotone with constant
if
Definition 2.3. ( [4] ). For any
and
, the Jordan product of p-order cone is expressed by
where
, and
,
.
Definition 2.4. ( [16] ). Let
. Then, the projection of x onto
is defined as
(5)
where
,
with
Definition 2.5. ( [16] ). Let
, then x can be decomposed into
where
and
if
. If
, then any vector in
satisfying
.
Lemma 2.1. ( [5] ). If the projection formula is defined as (2), then the properties below can be received:
(i)
(ii)
(iii)
(iv)
3. Sample Average Approximation
If the integral involved in POSVI can be evaluated, then it can be solved as a deterministic stochastic variational inequality problem. However,
are usually not accurately evaluated, because the distribution of
is unknown and the information of
can only be acquired from the past data samples, which inspires us to search a function to approximate
. Many scholars have explored approximation methods [17] [18] [19]: sample-path optimization (SPO), sample average approximation (SAA) and stochastic approximation (SA). In this paper, we select SAA method, whose main idea is to generate N independent and identically distributed (i.i.d.) samples
, and use the sample average function
to approximate
. Then we acquire the problem: find
, such that
(6)
where
, we call (1) as true problem and (6) as SAA problem.
Definition 3.1 ( [20] ) Let
denote jth component of
,
. Define the moment function of
as:
andthe moment function of
is
thus,
Lemma 3.1. ( [20] ) Let X be a compact subset of C,
. Suppose the conditions hold below:
1) The moment function
is finite with respect to (w.r.t.) x in a certain neighborhood of zero;
2)There is a metric function
,such that for any
,and
, there is
3) The moment function
of
is finite w.r.t. t in a certain neighborhood of zero.
Then for any
, there exists
,
,independent of N, such that
Lemma 3.1 guarantees that the approximation by the SAA method is reasonable.
Lemma 3.2. ( [21] ). Let
be a solution of SAA problem (6) and
be set of solutions to true problem (1). Suppose that:
a) Lemma 3.1 holds;
b)
(
) exists, for every
and
;
Then for every
, there is
,
, independent of N, such that
w.p.1 (7)
forN sufficiently large.
denotes the distance from point x to set D.
Lemma 3.2 studies the optimal solution set of SAA problem (6) convergences to optimal solution of true problem (1) with probability one (w.p.1.). Let
again, according to (4) and (3), solving SAA problem (6) is equivalent to find a zero of
.
4. Projection and Contraction Method and Convergence Analysis
In this part, we propose the projection and contraction method based on the former research and certify the algorithm is convergent. At last, the algorithm is applied to numerical examples that we give. To facilitate our research, we denote the projection of x onto
as
and the projection of
onto
as
. Hence, it is clear that
for any
.
Algorithm 3.1.
Step 0 Given
, let
,
,
,
, and
denotes an arbitrary initial iteration point, Set
.
Step 1 Compute
, if
, stop; Otherwise, go to step 2.
Step 2 Let
be the smallest nonnegative integer, which satisfies
, such that
(8)
if
(9)
then
.
Step 3 Calculate
(10)
with
(11)
Step 4 Compute
(12)
Set
; go to step 1.
Next, the convergence of Algorithm 3.1 is researched.
Lemma 4.1 Suppose that
is a solution of the SAA problem (6),
and
are respectively defined by (10) and (11). Then, the inequality holds
Proof. From the definition of projection in formula (2), we know
, obviously the lemma holds.
Lemma 4.2. Let
and
, the following formula holds:
Proof. Prove the above inequality is equivalent to proof
(13)
From Formula (2), we can get
(14)
then from Lemma 2.1 (1) and (13), we get
(15)
thus we have
(16)
Let
and
, the formula holds
(17)
the above formula holds because of Lemma 2.1. So the inequality (16) and (17) hold, the proof is complete.
Theorem 4.1. Assume that
is monotonous and Lipschitz continuous w.r.t. x, and constant
. Let the solution set of SAA problem (6) be nonempty. Then the sequence
obtained by Algorithm 3.1 converges to the solution of (6).
Proof. From (10) and (11), it gets that
(18)
(19)
which is equivalent to
that is
the above inequality holds since the function
is Lipschitz continuous. Let
be a solution of SAA problem (6), then
the first inequality holds since the projection operator is non-expansive, the second inequality holds since Lemma 4.1, and the last inequality holds from inequality (18) and (19). Obviously, the sequence
is bounded.
Let
be a cluster point of
and
is a subsequence of
, which converges to
. Denote
, since
is continuous, then
Hence,
is a solution of (6).
Next, it proves that the sequence
has only one cluster point. Suppose that there is
such that
converges to
, and denote
since
is a cluster point of the sequence
, there is a
such that
thus, it follows that
and
which contradicts the assumption. Thus it is certified completely, and the sequence
converges to the solution of SAA problem (6).
5. Numerical Experiment
In this section, we certify the effectiveness of algorithm 3.1 by giving some numerical experiments. The tasks are completed by Matlab 2018b, which are installed on a computer that has 3.3 GHz CPU and 4.0 GB memory. We set parameters
respectively, and
denotes error defined by
, further, ITER represents average number of iterations and ACPU is the average run time.
Example 5.1. Consider the POSVI below: find x, such that
where
,
, D is a real symmetric matrix with n dimension, and the element of D is selected randomly by Matlab from the interval
. The SAA problem: find
, such that
We choose
respectively, and the corresponding numerical results are exhibited in Tables 1-4.
Table 1. Solving numerical results when
.
Table 2. Solving numerical results when
.
Table 3. Solving numerical results when
.
The error
corresponding to every p we choose changes with time as shown in Figures 1-4.
Table 4. Solving numerical results when
.
From Tables 1-4 and Figures 1-4, we can analyze that for different p, as the dimension n increases, ITER and ACPU both become larger and relatively stable, and the error
becomes smaller and tends to zero. Therefore, generally speaking, the algorithm we proposed is effective.
Specially, p = 2 is a special case. When p = 2, P-order cone degenerates into second order cone. From the numerical results, when p = 2 and p = 3, the operation result is better when n is larger. 3 is the smallest number greater than 2. After many numerical experiments, it is found that with the increase of p, the time for large-scale problems will become longer and longer, until p = 10, the calculation results tend to be stable. That is, after p is greater than 10, the calculation result does not change much. So we chose a middle number, 5.
6. Conclusion
In this paper, a sample average approximation method is applied to approximate stochastic inequality problem with p-order cone. In order to solve p-order cone stochastic variational inequality problem, under moderate condition, we propose a projection and contraction method and prove the iteration sequence produced by the method converges to the solution of the SAA problem. At last, based on the projection and contraction method, we give some numerical examples and the experimental results verify the accuracy of the method. Next, we expect to investigate the numerical solution of circular cone constrained stochastic variational inequality problem. Find other suitable methods to further solve the p-order cone-constrained stochastic variational inequality problem in this paper, and carry out numerical simulation comparison.