Uncertain convex programs: randomized solutions and confidence levels

By Giuseppe Carlo Calafiore, Marco Claudio Campi. Mathematical programming, 2005

779 citations recorded by the source corpus.

Subfields: Management Science and Operations Research, Ocean Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7180f301c27c4cb7ce0e0b759b34fe9eba5ea40f

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Citations in the corpus

A tractable approximation of non-convex chance constrained optimization with non-Gaussian uncertainties (2015, methodology): “Scenario generation methods (Califore and Campi 2005, 2010) are applicable irrespective of the type of distribution function of the uncertain variables (Gaussian or non-Gaussian).”
On Ensuring that Intelligent Machines Are Well-Behaved (2017, methodology): “…constraints to hold for all P ∈ P. Scenario approximation methods can apply when P , the distribution of Wi, is neither known nor within a known set, P, but samples (realizations of Wi) can be generated from P [De Farias and Van Roy, 2004; Calafiore and Campi, 2005; Nemirovski and Shapiro, 2006].”
Preventing undesirable behavior of intelligent machines (2019, background): “Scenario approximation methods can apply when P is neither known nor within a known set, but samples (realizations of Wi) can be generated from P [79, 80, 81].”
An integer programming approach for linear programs with probabilistic constraints (2007, background): “Examples of these conservative approximations include scenario approximation [6, 17] and robust optimization, e.”
The scenario approach to robust control design (2006, methodology): “The same authors of the present paper have published a previous contribution in [24], dealing with convex optimization within a probabilistic framework.”
The Stochastic Vehicle Routing Problem for Minimum Unmet Demand (2009, neutral): “the link between the chance-constrained technique and robust optimization [32, 12, 19, 14].”
Monte Carlo sampling-based methods for stochastic optimization (2014, result): “Still, Campi et al. [38] improve upon an original result by Calafiore and Campi [36] and show that, given 0 < δ ≤ 1, by choosing N ≥ 2 α log 1 δ + d x (recall that d x is the dimension of x in (SP)), the optimal solution to (42) is feasible to the original problem (40) with probability at least 1 − ”
Optimization under uncertainty with applications to design of truss structures (2008, background): “This scenario approximation of the worst-case problem has been first introduced in Calafiore and Campi (2005), where it has been shown that the solution obtained from the scenario problem is actually approximately feasible for the original worst-case problem, in a sense explained next.”
Optimization under Uncertainty in the Era of Big Data and Deep Learning: When Machine Learning Meets Mathematical Programming (2019, background): “The proof of this theorem depends on the fundamental fact that the number of support constraints, the removal of which changes the optimal solution, is upper bounded by the number of decision variables [165].”
A Robust Optimization Perspective on Stochastic Programming (2007, background): “Computational tractability of robust linear constraints is achieved by considering tractable uncertainty sets such as ellipsoids (see Ben-Tal and Nemirovski 2000) and polytopes (see Bertsimas and Sim 2004a), which yield robust counterparts that are second-order conic constraints and linear…”
A practical guide to robust optimization (2015, background): “Alternatively, if the probability distribution of the uncertain parameter is known, one may also use the randomized sampling of the uncertainty set proposed by Calafiore and Campi (2005).”
On Complexity of Stochastic Programming Problems (2005, background): “This leads to the chance, or probabilistic, constraints formulation which is going back to Charnes and Cooper [11].”
Ambiguous chance constrained problems and robust optimization (2006, background): “Calafiore and Campi [10] [11] consider general convex functions f (x, h) and show that for N ≥ 2n ln 12 + 2 ln 2 δ + 2n, the optimal solution of the sampled problem (5) is feasible for (3) with probability at least 1−δ.”
Theory and Applications of Robust Optimization (2010, background): “Many approaches have been considered to address the computational challenges they pose [121, 112], including work using sampling to approximate the chance constraints [46, 47, 76].”