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Risk and complexity in scenario optimization

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Abstract

Scenario optimization is a broad methodology to perform optimization based on empirical knowledge. One collects previous cases, called “scenarios”, for the set-up in which optimization is being performed, and makes a decision that is optimal for the cases that have been collected. For convex optimization, a solid theory has been developed that provides guarantees of performance, and constraint satisfaction, of the scenario solution. In this paper, we open a new direction of investigation: the risk that a performance is not achieved, or that constraints are violated, is studied jointly with the complexity (as precisely defined in the paper) of the solution. It is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk. This result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios; in particular, no extra knowledge on the distribution by which scenarios are generated is assumed, so that the result is broadly applicable. This deep-seated result unveils a fundamental and general structure of data-driven optimization and suggests practical approaches for risk assessment.

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Notes

  1. No limitations are imposed on \({\varDelta }\) like e.g. that \({\varDelta }\) is a subset of a Eucledian space or of a vector space, nor is \({\varDelta }\) endowed with a metric or a topology. \({\varDelta }\) is just a generic set that forms a probability space together with \(\mathcal {F}\) and \({\mathbb {P}}\). Hence, ideas like “the sample \(\delta _i\), \(i=1,\ldots ,N\), covers, or fills up, \({\varDelta }\)” are void of any meaning. This generality in the definition of \({\varDelta }\) is important for the widespread applicability of the theory.

  2. We assume that function \(\ell (v,\delta )\) is convex in v for any given value of \(\delta \), while its dependence on \(\delta \) is arbitrary.

  3. Throughout, we assume that a solution exists. If more than one solution exists, a solution is singled out by means of a convex tie-break rule according to the approach of [10].

  4. We remark that \(V(x^*_N)\) quantifies the risk, which refers to the chance-constrained feasibility, while the value is not at issue here.

  5. Similarly to problem (1), it is assumed that the problems obtained after removing one constraint from (1) admit a unique solution, possibly after breaking the tie by means of a convex tie-break rule according to the approach of [10].

  6. To this purpose, it is enough to eliminate one by one the constraints and recompute the solution, the support constraints are those whose elimination determines a change in the solution.

  7. These are the same empirical distributions as in Figs. 3 and 4 (in Figs. 3 and 4 only some values of k were displayed).

  8. \({{\mathcal {Z}}}\) can be any set, without any Euclidean structure. We change notation from \({{\mathcal {X}}}\) to \({{\mathcal {Z}}}\) because in some applications \({{\mathcal {Z}}}\) is the same as \({{\mathcal {X}}}\) augmented with extra elements; concrete examples of decision sets are provided in Sects. 5.1 and 5.2.

  9. Note that only the tie with respect to x is broken by \(t_1(x)\), \(t_2(x)\), \(t_3(x)\), .... On the other hand, for a given \(x^*_m\) the values of \(\xi _i\), \(i=1,\ldots ,m\), remain unambiguously determined at optimum by relation \(\xi ^*_{i,m} = f(x^*_m,\delta _i)\), so that no tie on \(\xi _i\), \(i=1,\ldots ,m\), can persist after the tie on x is broken.

  10. Intuitively, the proportion of violated constraints (empirical risk) is not a valid indicator of the true risk \(V(x^*_N)\) since optimization generates a bias towards larger risks by drifting the solution against the constraints. The excess with respect to the number of violated constraints that appears in the computation of \({\tilde{s}}^*_N\) captures this mechanism and offers one the possibility to obtain tight evaluations of the risk, as quantified by the lower and upper bounds in Eq. (19), independently of the problem under consideration.

  11. The increase is not always monotone.

  12. Other methods have been proposed in the literature to trade the risk for an improved cost. One method consists in allowing the solution to violate a preset proportion of the empirical constraints (chance-constrained problem over the empirical distribution). In the context of scenario optimization, this approach is described in [18], where practically useful, but untight, bounds on the risk are also derived. More generally, the problem of relating the empirical risk to chance-constrained feasibility is dealt with in many papers including [6, 7, 44, 57]. The problem of finding a solution that violates a preset proportion of the empirical constraints is a non-convex problem that is difficult to solve in general. The formulation in (14) is convex and this eases the problem of finding a solution. Interestingly, as already noted, this formulation is amenable to tight evaluations of the risk.

  13. For reproducibility, we inform the reader about the mechanism by which \(q_{j,k}(\delta )\) were generated. Let \(\delta =(\alpha _1,\alpha _2,\gamma _{1,1},\ldots ,\gamma _{50,1},\gamma _{1,2},\ldots ,\gamma _{50,2})\), where, for \(k=1,2\) and \(j=1,\ldots ,50\), \(\alpha _k \sim \mathcal {U}[10,50]\) (i.e., \(\alpha _k\) is uniformly distributed in [10, 50]), \(\gamma _{j,k} \sim {\mathcal {U}}[-2.5,2.5]\), and all these variables are independent one of the others. Then, \(q_{j,k}(\delta ) = (\alpha _k^{\frac{1}{4}} + \gamma _{j,k})^{-1}\), \(j=1,\ldots ,50\), \(k=1,2\). Moreover, in the simulation, we took \(a_1=a_2=1\).

  14. Notice that, strictly speaking, this choice of \(f(x,\delta )\) does not satisfy Assumption 6. Reason is that setting to zero \(f(x,\delta )\) when \(\sum _{j=1}^{50} q_{j,1}(\delta ) x_j - a_1\) and \(\sum _{j=1}^{50} q_{j,2}(\delta ) x_j - a_2\) are negative, as is done in (22), generates regions with positive volume in the domain in \( { {{\mathbb {R}}}^{50} } \) for x where \(f(x,\delta ) = 0\). However, an easy inspection of the derivation of Theorem 4 shows that the requirement of Assumption 6 that, for every x, \({\mathbb {P}}\{\delta : f(x,\delta ) = 0\} = 0\) can be relaxed to requiring that, for every x, \({\mathbb {P}}\{\delta : x \text{ is } \text{ on } \text{ the } \text{ boundary } \text{ of } \text{ the } \text{ constraint } \{f(x,\delta ) \le 0 \} \} = 0\), and the theory goes through unaltered with the only modifications that, throughout, “\(f(x,\delta ) = 0\)” becomes “x is on the boundary of the constraint \(\{f(x,\delta ) \le 0 \}\)”, “\(f(x,\delta ) < 0\)” becomes “x is in the interior of the constraint \(\{f(x,\delta ) \le 0 \}\)”, and “\(f(x,\delta ) \ge 0\)” becomes “x violates or is on the boundary of the constraint \(\{f(x,\delta ) \le 0 \}\)”. While we have preferred in the general presentation the simpler formulation of Assumption 6, this second formulation leads to zero volume regions in the domain in \( { {{\mathbb {R}}}^{50} } \) for x in the present example.

  15. Since the intervals in Fig. 12 are obtained by a repeated application of Theorem 4, the confidence that \({\underline{\epsilon }}({\tilde{s}}^*_{2000}) \le V(x^*_{2000}) \le {\overline{\epsilon }}({\tilde{s}}^*_{2000})\) for all the 22 values of \(\rho \) simultaneously is \(1-22\cdot \beta \).

  16. The reason for introducing H is that the theorem will be proven in a slightly more general form where H is any integer \(\ge 1\) and not just 3N. The choice \(H = 3N\) gives satisfactory evaluations in most cases, and this is why Theorem 2 was stated with \(H = 3N\). However, the extra generality allowed by other values of H can turn out to be useful to tighten the bounds \({\underline{\epsilon }}(\cdot )\) and \({\overline{\epsilon }}(\cdot )\) in some cases when N is not too large. This issue is not further discussed in this paper.

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  1. Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    S. Garatti

  2. Dipartimento di Ingegneria dell’Informazione, Università di Brescia, Via Branze 38, 25123, Brescia, Italy

    M. C. Campi

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MATLAB code

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The following MATLAB code returns \({\underline{\epsilon }}(k)\) and \({\overline{\epsilon }}(k)\) for user assigned k, N, and \(\beta \).

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Garatti, S., Campi, M.C. Risk and complexity in scenario optimization. Math. Program. 191, 243–279 (2022). https://doi.org/10.1007/s10107-019-01446-4

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