Abstract
In this paper, we mainly consider the stochastic second-order cone complementarity problem (SSOCCP). Due to the existence of stochastic variable, the SSOCCP may have no solutions. In order to deal with this problem, we first regard the merit function of the stochastic second-order cone complementarity problem as the loss function and then present a low-risk deterministic model that is a conditional value-at-risk (CVaR) model. However, there may be two difficulties for solving the CVaR model directly: One is that the objective function is a non-smoothing function. The other is that the objective function contains expectation. (In general, the value of expectation is not easy to be calculated.) In view of these two problems, we present the approximation problems of the model by using a smoothing method and a sample average approximation technique. Furthermore, we give the convergence results of global optimal solutions and the convergence results of stationary points of the approximation problems, respectively.
Highlights
The second-order cone in Rn is defined asKn = (x1, x2) ∈ R × Rn–1| x2 ≤ x1, where · denotes the Euclid norm
The KKT condition of second-order cone programming (SOCP) problem can be equivalent to a second-order cone complementarity problem
Fukushima et al [10] proved that the min function and the FB function in NCP can be spread to Second-order cone complementarity problem (SOCCP) by using Jordan algebra
Summary
Second-order cone complementarity problem (SOCCP) is as follows: Find a vector x ∈ We consider a Fischer–Burimister(FB) second-order cone complementarity function, taking advantage of the knowledge about Jordan algebra, φFB(x, y) can be written as φFB(x, y) = x + y – λ1u1 + λ2u2 .
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