Input parameters of optimization problems are often affected by uncertainties. Chance constraints are a common way to model stochastic uncertainties in the constraints. We consider joint chance-constrained optimization problems with continuous distributions and non-convex constraint functions. For this difficult class of problems, no efficient general solution approaches exist. We propose a gradient-based approach to approximately solve the joint chance-constrained problems. Therefore, we approximate the original problem by smoothing indicator functions and penalize a violation of the smoothed constraint in the objective function. The approximation problem is solved with the Continuous Stochastic Gradient method, a variant of the stochastic gradient descent with improved convergence properties that has recently been introduced in the literature. Furthermore, we provide a convergence theory for the smoothing and penalty approximations with sequences of increasing smoothing and penalty parameters. Under very mild assumptions, our approach is applicable to a wide range of joint chance-constrained optimization problems. We illustrate the efficiency of our approach for difficult practical problems with an application to gas networks. The numerical experiments demonstrate that the approach quickly finds nearly feasible solutions for difficult joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances.

Chance constraints naturally emerge in systems where reducing risk is crucial, particularly those exposed to high-impact but rare events such as natural disasters, infrastructure failures, and financial crashes. However, the rarity of such events renders the solution of chance-constrained models impractical using naive Monte Carlo methods. Importance sampling can be a viable solution method in such cases, as it employs an alternate probability distribution to generate samples more frequently from rare-event regions. Nonetheless, the challenge of selecting an appropriate alternate distribution that is effective for all decision variables across various problems remains unresolved. Most existing strategies are problem-specific, necessitating an in-depth understanding of the system's dynamics. Our work adapts the self-structured importance sampling (SSIS) method, recently proposed by Deo \&; Murthy, to develop a new algorithm for solving chance-constrained problems. This method transforms samples generated through naive Monte Carlo to samples that more accurately represent rare region behavior and can be universally applied across a wide range of constraint functions and probability distributions. Our results demonstrate that the proposed algorithm can substantially outperform Monte Carlo sampling in terms of both feasibility and optimality, even with small sample sizes.

Designing controllers for contact-rich systems is known to be challenging due to the non-smooth dynamics. In general, the parameters involved in contact-rich systems such as friction coefficients are very difficult to estimate perfectly. This can lead to poor performance during execution. However, considering uncertainties for contact-rich systems is even more challenging and very little understood in open literature. In this talk, we talk about novel insight regarding stochastic contact-rich systems and present a novel contact-aware covariance controller for the same. The proposed formulation highlights a very fundamental treatment of stochastic contact-rich systems using sample average approximation. Furthermore, a novel non-linear programming method using bilevel optimization is proposed, resulting in much better computation compared to the mixed-integer optimization-based stochastic optimization, which has been used in the covariance steering community. This paper makes an important contribution toward safe manipulation and locomotion in the real world using stochastic contact-rich models.