Research problem:
Tight outage probability control in probabilistically/chance constrained
optimization problems
Brief Description:
In many applications, some of the
variables in an optimization problem might contain uncertainty. If a constraint
depends on a variable with uncertainty, the constraint is not deterministic
anymore. One example is in wireless communications, in which the constraint
might depend on the channel state information (CSI). If the CSI contains
uncertainty, the constraints become random. One possible way to handle
non-deterministic constraint is to replace it by its averaged form. However,
such replacement would lead to the constraint not being satisfied half of the
time. A better way is to impose an allowable outage probability for the
constraint (e.g., allowing the constraint not being satisfied 5% of the time).
Unfortunately, such probabilistic constraint is challenging to handle, as
usually there is no closed-form expression for the constraint. A common get
around is to replace the probabilistic constraints with a stricter constraint
but not in probabilistic form (usually Bernstein-type inequality is involved
here). While this would end up a feasible solution for the original problem,
the realized constraint is usually over-satisfied and a better solution might
be missed as the feasible set is shrunk in exchange for easy handling of the constraint.
The aim of our study in this topic is to control
the outage probability as precise as possible, so that it allows for better
exploration of the feasible set. To this end, we explored three strategies.
Firstly, we proposed a general “set-squeezing procedure” for
handling a wide range of continuous uncertainty, in which in each iteration,
the largest convex set around the current solution is located and a new
solution is searched. This allows efficient exploration of different parts of a
non-convex feasible set, and guarantees a local optimum or a boundary solution
under mild conditions. Theoretical proof of convergence is provided for the
following classes of uncertainties and constraints:
·
Probabilistic
signal-to-interference plus noise (SINR) constraint under a general class of
quadratic continuous uncertainty (which covers Gaussian, Laplace,
t-distribution, and ellipsoid with bounded support) [1]
·
Probabilistic
mean-square error (MSE) constraint under arbitrarily distributed uncertainty
with known mean and covariance [2]
·
Constraint
being a quadratic function under a known uncertainty probability density
function [3]
Applications to wireless transceiver design
problems show that the proposed set-squeezing procedure could tightly realize
the outage constraint, and consequently a better objective function value is
obtained compared to Bernstein-type inequality approaches.
Secondly, for Gaussian distributed
uncertainty, there exists closed-form expression for the probabilistic
constraint if it is related to MSE or SINR. It
turns out that when MSE or SINR is rewritten in a quadratic form with respect
to the uncertainty, various strategies can be exploited to equivalently
transform the probabilistic constraint into a deterministic one. In this
aspect, the following specific scenarios were studied:
·
If
the quadratic form is positive semidefinite, the exponential distribution
property of the uncertainty power is sufficient
to convert the probabilistic constraint into a closed-form deterministic
constraint [4][5];
·
If
the quadratic form is indefinite, a tighter constraint with a tuneable
parameter can be further applied, and the
tunning parameter can be used to control the tightness of probabilistic
constraint realization [6];
·
If
the MSE or SINR expression appears in the form
of the determinant of a Hermitian matrix (in multiple antennas cases), the
quadratic form can be obtained by leveraging the properties of rank equalities
and inequalities, and properties of gamma
distribution [7].
Thirdly, in certain cases, even if the probabilistic constraint can be transformed into a deterministic form, it is still complicatedly related to other optimization variables, which might require further relaxation or approximation. We employed the concept of implicit function to get rid of the approximation and relaxation. This idea has been demonstrated in the context of collaborative eavesdroppers [8] and secure integrated sensing and communications (ISAC) [9].
Related
Publications:
[2]
Xin He and Yik-Chung Wu, ``Probabilistic QoS Constrained Robust Downlink
Multiuser MIMO Transceiver Design with Arbitrarily Distributed Channel
Uncertainty,"
IEEE Trans. on Wireless Communications, Vol. 12, no.
12, pp.6292-6302, Dec 2013.
[3] Xin He
and Yik-Chung Wu, ``Set Squeezing
Procedure for Quadratically Perturbed Chance-constrained Programming," in IEEE
Trans. on Signal Processing, vol. 69,
pp. 682-694, 2021, doi: 10.1109/TSP.2020.3047200.
[4] Zongze
Li, Shuai Wang, Miaowen Wen, and Yik-Chung Wu, ``Secure Multicast Energy-Efficiency Maximization
with Massive RISs and Uncertain CSI: First-order Algorithms and Convergence
Analysis," IEEE Trans. on
Wireless Communications,
vol. 21, no. 9, pp. 6818-6833, Sep 2022.
[5] Zongze
Li, Shuai Wang, Pengcheng Mu, and Yik-Chung Wu, ``Probabilistic
Constrained Secure Transmission: Variable-Rate Design and Performance
Analysis," IEEE Trans. on
Wireless Communications, Vol. 19, no. 4, pp.
2543-2557, April 2020.
[6] Zongze
Li, Minghua Xia, Miaowen Wen, and Yik-Chung Wu, ``Massive Access in
Secure NOMA under Imperfect CSI: Security Guaranteed Sum-rate Maximization with
First-order Algorithm," IEEE
Journal on Selected Areas in Communications (JSAC), vol. 39, no. 4, pp. 998-1014, Apr. 2021.
[7] Zongze Li, Qingfeng Lin, Yik-Chung Wu, Derrick Wing Kwan Ng, and Arumugam Nallanathan, ``Enhancing Physical Layer Security with RIS under Multi-Antenna Eavesdroppers and Spatially Correlated Channel Uncertainties," IEEE Trans. on Communications, vol. 72, no. 3, pp. 1532-1547, Mar 2024.
[8] Hancheng Zhu, Zongze Li, Yik-Chung Wu, and H. Vincent Poor, ``Countering Collaborative Eavesdroppers under Imperfect CSI: Outage Probability Constraint Transformation and Zeroth-Order Optimization," accepted in IEEE Trans. on Signal Processing, 2026
[9] Hancheng Zhu, Zongze Li, and Yik-Chung Wu,
``Unified framework for outage-constrained rate maximization in secure ISAC
under various sensing metrics," accepted in IEEE Journal on Selected
Areas in Communications (JSAC), 2026