Robust solutions to box-constrained stochastic linear variational inequality problem

We present a new method for solving the box-constrained stochastic linear variational inequality problem with three special types of uncertainty sets. Most previous methods, such as the expected value and expected residual minimization, need the probability distribution information of the stochastic variables. In contrast, we give the robust reformulation and reformulate the problem as a quadratically constrained quadratic program or convex program with a conic quadratic inequality quadratic program, which is tractable in optimization theory.


Introduction
Variational inequality theory is an important branch in operations research. Recall that the variational inequality problem, denoted by VI(X, F), is to find a vector x * ∈ X such that where X ⊆ R n is a non-empty closed convex set, F : X → R n is a given function. A particularly important class of VI(X, F) is the box-constrained variational inequality problem (see []), denoted by VI(l, u, F), where X = D = {x ∈ R n | l ≤ x ≤ u}, l := (l  , . . . , l n ) T , u := (u  , . . . , u n ) T , along with the lower bounds l i ∈ R ∪ {-∞}, the upper bounds u i ∈ R ∪ {+∞} and l i < u i for all i = , . . . , n. To obtain the solutions of VI(l, u, F), many methods are presented based on the KKT system, which is given as follows: () Note that this KKT system is a complementarity problem in fact.
The box-constrained variational inequality problem has many applications ranging from operations research to economic equilibrium and engineering problems []. However, some elements may involve uncertain data in practice. For example, the demands are generally difficult to be determined in supply chain network, because they vary with the change of income level [, ]. Moreover, in traffic equilibrium problems, the selfish users' attempt to minimize travel cost leads the equilibrium (or steady-state) flows to uncertainty For the existence of stochastic elements, the solutions of stochastic variational inequality problem would change with stochastic elements respectively. In order to meet the needs in practice, many researchers begin to consider the following stochastic variational inequality problem, denoted by SVI(X, F), which requires an x * such that where ω ∈ ⊆ R τ is a stochastic vector and a.s. is the abbreviation for 'almost surely' under the given probability measure.
where t = xl ∈ R n + and t y ∈ R n + are vectors, I denotes an n-dimensional identify matrix and  represents a zero matrix with suitable dimension.
Before proceeding, we briefly touch upon earlier efforts about SVI(X, F). For example, the expected value (EV) method [-] and the expected residual minimization (ERM) method [-] focused on minimizing the average or the average of expected residual.
These methods needed the information of a probability distribution and focused on providing estimators of local solutions. In the spirit of robust approaches, instead of ERM or EV, we consider the minimization of the worst-case residual over a particular uncertainty set . By employing KKT system (), we give the robust reformulation of SLVI(l, u, F) as follows: Obviously, () can be rewritten as follows: Note that t y ≥  solves () if and only if t y is a solution of () with optimal value zero by Lemma . in [].
The organization of this paper is as follows. In Section , we discuss the tractable robust counterparts of monotone SLVI(l, u, F) in different uncertain sets. Non-monotone generalizations and their tractable robust counterparts form the core of Section . In Section , we give the conclusions.

Tractable robust counterparts of monotone SLVI(l, u, F)
To begin with, we provide robust counterparts in regimes where M(ω  ) is a stochastic positive semidefinite matrix and q(ω  ) is a stochastic vector with ω  ∈  and ω  ∈  , where  and  are uncertainty sets.  contains  ∞ and   . These two types of uncertainty sets state as follows:  contains  ∞ ,   and   , these three types of uncertainty sets state as follows: Here, · ∞ , ·  , ·  denotes infinite norm, -norm and -norm, respectively. Moreover, we define M( . . , S, is an n-dimensional symmetric positive semidefinite matrix and q s , s = , . . . , S, is an ndimensional vector. Without loss of generality, we assume that M(ω  ) is symmetric; if not, we may replace the matrixes by their symmetrized counterparts. This assumption guarantees M(ω) I ). In this section, we first consider the case when  =  ∞ , the robust counterpart of SLVI(l, u, F), and then focus on the case  =   while ω  ∈  . We now prove that robust problem () can be reformulated as a quadratically constrained quadratic program (QCQP) or convex program with a conic quadratic inequality quadratic program [, ] under the different uncertain sets.

Theorem  Consider the optimization problem
Then () can be reformulated respectively as QCQP or convex program with a conic quadratic inequality quadratic program while ω  belongs to different sets of  .
Proof Case :  =  ∞ . We first give the equivalent reformulation of the first constraint in problem () as follows: Noting that η  = max ω ∞ ≤ η T ω, we evaluate the maximum in the above formulation as follows: where the last equality is obtained by applying the positive semidefiniteness of M s for s = , . . . , S. Similarly, we have Hence, reformulation () can be rewritten as: We then give the equivalent form of the second constraint in problem () as follows: After a simple calculation, for i = , . . . , n, s = , . . . , S, we have By the fact that the upper bound u ≥ x, we simplify formulation () as follows: Here, i = , . . . , n and for a given vector x ∈ R n , |x| = (|x  |, . . . , |x n |) T . Then problem () can be rewritten as follows: Since it is difficult to compute maximum and absolute value functions, we can eliminate them by increasing relaxation variables a s ∈ R + , α s , β s ∈ R n + , s = , . . . , S. Then the above problem can be converted to QCQP: We first replace ω  s ∈  ∞ in formulation () by ω  s ∈   and noting that η ∞ = max ω  ≤ ω T η, then for i = , . . . , n, we have After a simple calculation, problem () can be rewritten as follows: In order to calculate easily, we introduce variables a ∈ R + , α s , β s ∈ R n + , s = , . . . , S. Then we obtain the following QCQP: a, t, y ≥ .
We now consider the robust counterpart of () defined by  =   and ω  ∈  .

Theorem  Consider the optimization problem ()
where ω  ∈   and M(ω  ) = M  + S s= ω  s M s , q(ω  ) = q  + S s= ω  s q s . Then () can be reformulated as QCQP or convex program with a conic quadratic inequality quadratic program while ω  belongs to different sets of  .
Proof Case :  =  ∞ . We first consider the equivalent form of quadratic constraint in problem (), it can be represented as We then have to evaluate the result of maximum in the above formulation by It then follows from () that formulation () can be rewritten as We then give the equivalent component form of the second constraint in problem () by formulation () as follows: Finally, by adding variables a s ∈ R + , s = , . . . , S, α, β ∈ R n + , we obtain QCQP as follows: -a s ≤ t T q s ≤ a s , Inspired by Case , we derive the conclusions of Case  and Case . Case :  =   . We replace ω  s ∈  ∞ in formulation () by ω  s ∈   and combining with (), (), () and (), by adding variables a ∈ R + , α, β ∈ R n + , problem () can be reformulated as the following QCQP: Case :  =   . We use ω  s ∈   instead of ω  s ∈  ∞ in formulation () and taking (), (), () and () into account, by adding variables a ∈ R + , α, β ∈ R n + , problem () can be converted to convex program with a conic quadratic inequality quadratic program as follows: These complete the proof.
Theorems  and  present an approach to seek robust solutions of SLVI(l, u, F), when M s (s = , . . . , S) is a positive semidefinite matrix. However, this condition is a little strict so that it is difficult to satisfy in practice. Thus, we give more general circumstances in Section .

Tractable robust counterparts of non-monotone SLVI(l, u, F)
In is no longer monotone for ω ∈ , a.s. In this section, we consider the case that q is a certain vector. Since it is difficult to directly apply the results of quadratic program, these problems are somewhat more challenging. As we proceed, we may still obtain the results that robust problem () can be reformulated as QCQP or convex program under suitably defined uncertain sets.
Theorem  Consider the optimization problem () with uncertain sets defined by  , () can be reformulated as QCQP or convex program.
Proof Case : =  ∞ . Firstly, we consider the first constraint in formulation (). Similar to (), (), combining with max ω ∞ ≤ η T ω = η  and M s , M k , for every s, k, we have In the same way, we have Then the first constraint in problem () can be transformed as follows: On the other hand, it follows from max ω  ≤ ω T η = η ∞ that, for every s, k, we can deduce that Finally, in order to eliminate the maximum function, we add extra variables α  , α  , β  , β  ∈ R n + into constraints. In addition, taking the fact h(x, ω) ≥ , ω ∈   , a.s. ⇔

Conclusions
We present the first attempt to give the robust reformulation for solving the boxconstrained stochastic linear variational inequality problem. For three types of uncertain variables, the robust reformulation of SLVI(l, u, F) can be solved as either a quadratically constrained quadratic program (QCQP) or a convex program, which are all more tractable and can provide solutions for SLVI(l, u, F), no matter for monotone or non-monotone F.