Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
Finding k points with minimum diameter and related problems
Journal of Algorithms
An efficient approach to removing geometric degeneracies
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On enclosing k points by a circle
Information Processing Letters
Mathematics of Operations Research
An introduction to support Vector Machines: and other kernel-based learning methods
An introduction to support Vector Machines: and other kernel-based learning methods
Robust Solutions to Uncertain Semidefinite Programs
SIAM Journal on Optimization
A Combinatorial Bound for Linear Programming and Related Problems
STACS '92 Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science
On Constraint Sampling in the Linear Programming Approach to Approximate Dynamic Programming
Mathematics of Operations Research
Uncertain convex programs: randomized solutions and confidence levels
Mathematical Programming: Series A and B
Probabilistically Constrained Linear Programs and Risk-Adjusted Controller Design
SIAM Journal on Optimization
Convex Approximations of Chance Constrained Programs
SIAM Journal on Optimization
A Sample Approximation Approach for Optimization with Probabilistic Constraints
SIAM Journal on Optimization
From Gestalt Theory to Image Analysis: A Probabilistic Approach
From Gestalt Theory to Image Analysis: A Probabilistic Approach
The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs
SIAM Journal on Optimization
Survey paper: Research on probabilistic methods for control system design
Automatica (Journal of IFAC)
A constraint sampling approach for multi-stage robust optimization
Automatica (Journal of IFAC)
Direct data-driven portfolio optimization with guaranteed shortfall probability
Automatica (Journal of IFAC)
Randomized sampling for large zero-sum games
Automatica (Journal of IFAC)
Stochastic model predictive control of LPV systems via scenario optimization
Automatica (Journal of IFAC)
Randomized Model Predictive Control for HVAC Systems
Proceedings of the 5th ACM Workshop on Embedded Systems For Energy-Efficient Buildings
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Random convex programs (RCPs) are convex optimization problems subject to a finite number $N$ of random constraints. The optimal objective value $J^*$ of an RCP is thus a random variable. We study the probability with which $J^*$ is no longer optimal if a further random constraint is added to the problem (violation probability, $V^*$). It turns out that this probability rapidly concentrates near zero as $N$ increases. We first develop a theory for RCPs, leading to explicit bounds on the upper tail probability of $V^*$. Then we extend the setup to the case of RCPs with $r$ a posteriori violated constraints (RCPVs): a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control. Explicit and nonasymptotic bounds are derived also in this case: the upper tail probability of $V^*$ is upper bounded by a multiple of a beta distribution, irrespective of the distribution on the random constraints. All results are derived under no feasibility assumptions on the problem. Further, the relation between RCPVs and chance-constrained problems (CCP) is explored, showing that the optimal objective $J^*$ of an RCPV with the generic constraint removal rule provides, with arbitrarily high probability, an upper bound on the optimal objective of a corresponding CCP. Moreover, whenever an optimal constraint removal rule is used in the RCPVs, then appropriate choices of $N$ and $r$ exist such that $J^*$ approximates arbitrarily well the objective of the CCP.