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Chance-Constrained Optimization with Complex Variables

Raneem Madani, Abdel Lisser

TL;DR

This work introduces Complex Chance-Constrained Programming (CCCP) for linear optimization with complex Gaussian uncertainties, addressing both individual and joint constraint settings. The individual CCCP is exactly reformulated as a deterministic Second-Order Cone Program (SOCP), leveraging the distribution of real parts of complex affine functions. For the joint CCCP, the authors develop copula-based dependency modeling and derive lower-bound (via Taylor expansion) and upper-bound (via piecewise linear) approximations that yield solvable SOCPs. Through adaptive MVDR beamforming with mismatch, the approach achieves superior performance and faster convergence compared with existing methods, and a detailed comparison shows the joint formulation outperforming the individual one. These results establish CCCP as a practical, interpretable framework for complex-valued stochastic optimization with strong implications for signal processing and related domains.

Abstract

Optimization problems involving complex variables, when solved, are typically transformed into real variables, often at the expense of convergence rate and interpretability. This paper introduces a novel formalism for a prominent problem in stochastic optimization involving complex random variables, termed the Complex Chance-Constrained Problem (CCCP). The study specifically examines the linear CCCP under complex normal distributions for two scenarios: one with individual probabilistic constraints and the other with joint probabilistic constraints. For the individual case, the core methodology reformulates the CCCP into a deterministic Second-Order Cone Programming (SOCP) problem, ensuring equivalence to the original CCCP. For the joint case, an approximation is achieved by deriving suitable upper and lower bounds, which also leads to a SOCP formulation. Finally, numerical experiments on a signal processing application, specifically the Minimum Variance Beamforming problem with mismatch using MVDR, demonstrate that the proposed formalism outperforms existing approaches in the literature. A comparative analysis between the joint and individual CCCP cases is also included.

Chance-Constrained Optimization with Complex Variables

TL;DR

This work introduces Complex Chance-Constrained Programming (CCCP) for linear optimization with complex Gaussian uncertainties, addressing both individual and joint constraint settings. The individual CCCP is exactly reformulated as a deterministic Second-Order Cone Program (SOCP), leveraging the distribution of real parts of complex affine functions. For the joint CCCP, the authors develop copula-based dependency modeling and derive lower-bound (via Taylor expansion) and upper-bound (via piecewise linear) approximations that yield solvable SOCPs. Through adaptive MVDR beamforming with mismatch, the approach achieves superior performance and faster convergence compared with existing methods, and a detailed comparison shows the joint formulation outperforming the individual one. These results establish CCCP as a practical, interpretable framework for complex-valued stochastic optimization with strong implications for signal processing and related domains.

Abstract

Optimization problems involving complex variables, when solved, are typically transformed into real variables, often at the expense of convergence rate and interpretability. This paper introduces a novel formalism for a prominent problem in stochastic optimization involving complex random variables, termed the Complex Chance-Constrained Problem (CCCP). The study specifically examines the linear CCCP under complex normal distributions for two scenarios: one with individual probabilistic constraints and the other with joint probabilistic constraints. For the individual case, the core methodology reformulates the CCCP into a deterministic Second-Order Cone Programming (SOCP) problem, ensuring equivalence to the original CCCP. For the joint case, an approximation is achieved by deriving suitable upper and lower bounds, which also leads to a SOCP formulation. Finally, numerical experiments on a signal processing application, specifically the Minimum Variance Beamforming problem with mismatch using MVDR, demonstrate that the proposed formalism outperforms existing approaches in the literature. A comparative analysis between the joint and individual CCCP cases is also included.

Paper Structure

This paper contains 14 sections, 6 theorems, 25 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex Chance Constraint Problem (CCCP)
  4. Normal Distribution of Complex Affine Functions
  5. Individual-CCCP
  6. Individual CCCP as SOCP
  7. Joint-CCCP
  8. Lower and Upper Approximations of Problem (P5)
  9. Lower Bound Approximation of (P5)
  10. Upper Bound - Piecewise Linear Approximation
  11. Simulation Experiments
  12. Adaptive Beamforming with Mismatch Problem
  13. Comparisons between individual and joint CCCP
  14. Conclusion

Key Result

lemma 1

If $c_j;j=1,\cdots,n$ are a random variables with $c_j\sim\mathcal{N}_c(\mu_{c_j}, \Gamma_{c_j}, C_{c_j})$, if the real and image parts of $c_j$ are independent, then $\Re(c^Hz)$ is real normally distributed with Furthermore, the variance is a quadratic function.

Figures (2)

  • Figure 1: SINR versus SNR for INR = 5 (left), INR = 20 (middle), and INR = 40 (right).
  • Figure 2: Output SINR with INR = $20$, $100$ runs and $\alpha = 0.7$. Problem was solved as an Individual CCCP and Joint CCCP.

Theorems & Definitions (13)

  • definition 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • ...and 3 more