Randomized Solutions to Convex Programs with Multiple Chance ConstraintsThis manuscript is the preprint of a paper submitted to the SIAM Journal on Optimization and it is subject to SIAM copyright. SIAM maintains the sole rights of distribution or publication of the work in all forms and media. If accepted, the copy of record will be available at http://www.siam.org.

Randomized Solutions to Convex Programs with Multiple Chance Constraintsthanks: This manuscript is the preprint of a paper submitted to the SIAM Journal on Optimization and it is subject to SIAM copyright. SIAM maintains the sole rights of distribution or publication of the work in all forms and media. If accepted, the copy of record will be available at http://www.siam.org.

Georg Schildbach Automatic Control Laboratory, Swiss Federal Institute of Technology, Zurich, Switzerland (schildbach|morari@control.ee.ethz.ch).    Lorenzo Fagiano Dipartimento di Automatica e Informatica, Polytecnico di Torino, Torino, Italy and Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara (CA), United States (lorenzo.fagiano@polito.it).    Manfred Morari
Abstract

The scenario-based optimization approach (‘scenario approach’) provides an intuitive way of approximating the solution to chance-constrained optimization programs, based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty (‘scenarios’). A key merit of this approach is that it neither requires explicit knowledge of the uncertainty set, as in robust optimization, nor of its probability distribution, as in stochastic optimization. The scenario approach is also computationally efficient because it only requires the solution to a convex optimization program, even if the original chance-constrained problem is non-convex. Recent research has obtained a rigorous foundation for the scenario approach, by establishing a direct link between the number of scenarios and bounds on the constraint violation probability. These bounds are tight in the general case of an uncertain optimization problem with a single chance constraint.

This paper shows that the bounds can be improved in situations where the chance constraints have a limited ‘support rank’, meaning that they leave a linear subspace unconstrained. Moreover, it shows that also a combination of multiple chance constraints, each with individual probability level, is admissible. As a consequence of these results, the number of scenarios can be reduced from that prescribed by the existing theory for problems with the indicated structural property. This leads to an improvement in the objective value and a reduction in the computational complexity of the scenario approach. The proposed extensions have many practical applications, in particular high-dimensional problems such as multi-stage uncertain decision problems or design problems of large-scale systems.

Key words: Uncertain Optimization, Chance Constraints, Randomized Methods, Convex Optimization, Scenario Approach, Multi-Stage Decision Problems.

1 Introduction

Optimization is ubiquitous in modern problems found in engineering, logistics, and other sciences. A common pattern is that a decision or design variable has to be selected from a subset of , as described by constraints , and its quality is measured against some objective or cost function :

(1.1a)
s.t. (1.1b)

1.1 Chance-Constrained Optimization

Unfortunately, in many practical applications the underlying problem data is uncertain. This uncertainty shall be represented with an abstract variable , where is an uncertainty set whose nature is not specified. The uncertainty may affect the objective function and/or the constraints . Thus for a particular decision it becomes uncertain what objective value is achieved and/or whether the constraints are indeed satisfied. The second situation represents a particular challenge, as good solutions are usually located on the boundary of the feasible set.

This gives rise to a trade-off problem between the (uncertain) objective value and the robustness of the chosen decision to a constraint violation. A large variety of approaches addressing this issue have been proposed in the areas of robust and stochastic optimization [3, 4, 5, 14, 15, 17, 19, 21], with the preferred method of choice depending on the requirements of the application at hand.

In many practical applications, can be assumed to be of a stochastic nature. In this case, the formulation of chance constraints, where the decision variable has to be feasible with a least probability for , has proven to be an appropriate concept for handling the uncertainty in the constraints. However, chance-constrained optimization problems are usually very difficult to solve. The scenario approach, as explained below, represents an attractive method for finding an ‘approximate solution’ to stochastic programs, since it is both intuitive and computationally efficient.

1.2 The Scenario Approach

Recent contributions [8, 10, 9, 11, 12] have revealed the theoretical links between the scenario approach and the solution to an optimization problem with a linear objective function and a single chance constraint ():

(1.2a)
s.t. (1.2b)

Here is a compact and convex set, denotes the transpose of a vector , is the probability measure on the uncertainty set , is a convex function in its first argument for -almost every uncertainty , and is some value in the open real interval .

The chance constraint (1.2b) is interpreted as follows. For any given , the left-hand side represents the probability of the event that indeed belongs to the feasible set. Written more properly,

(1.3)

however the left-hand side notation is kept throughout for brevity. Note that is considered to be a feasible point of the chance constraint (1.2b) if this probability is at least .

Remark 1.1 (Problem Formulation)

The formulation of the encompasses a vast range of problems, namely any uncertain optimization problem that becomes convex if the value of were fixed. (a) Any uncertain convex objective function can be included by an epigraph reformulation, with the new objective being a scalar and hence linear [7, Sec. 3.1.7]. (b) Joint chance constraints, where must satisfy multiple convex constraints simultaneously with probability , are covered since the intersection of convex sets is convex. (c) Additional deterministic, convex constraints can be included by intersection with the compact set .

The characterization of the feasible set of a chance constraint requires exact knowledge of the probability distribution of . Moreover, the feasible set is non-convex and difficult to express explicitly, except for very special cases [5, 14, 19, 21]. This makes the , in full generality and especially in higher dimensions , an extremely difficult problem to solve.

The scenario approach can be used to find an approximate solution to the , which is considered to be any point in that is feasible for the chance constraint with some given (very high) confidence . This problem is usually not as hard, if an approximate solution is chosen in a low-violation region of the decision space (with high confidence). However, then the resulting objective value may be poor, in which case the approximate solution shall be called ‘conservative’. Clearly, it is of major interest to find approximate solutions that are the least conservative (i.e. with an objective value as low as possible), and this is the goal of the scenario approach.

The basic idea of the scenario approach is to draw a specific number of samples (‘scenarios’) from the uncertainty , and to take the optimal solution that is feasible under all of these scenarios (‘scenario solution’) as an approximate solution. Computing the scenario solution involves a deterministic optimization program (‘scenario program’), which is obtained by replacing the chance constraint (1.2b) with the sampled deterministic constraints.

By construction, the scenario program is a deterministic, convex optimization program that can be solved efficiently by standard algorithms [7, 16, 18]. Moreover, the scenario approach is distribution-free in the sense that it does not rely on a particular mathematical model for the distribution of , or even its support set . In fact, both may be unknown; the only requirements are stated in the following assumption.

Assumption 1.2 (Uncertainty)

(a) The uncertainty is a random variable with (possibly unknown) probability measure and support set . (b) A sufficient number of independent random samples from can be obtained.

Note that Assumption 1.2 is fairly general. It could even be argued that the scenario approach is at the heart of any robust and stochastic optimization method, because either the uncertainty set or the probability distribution of are usually constructed based on some (necessarily finite) experience of the uncertainty.

Tight bounds for the proper choice of the sample size are established by [9, 11], when linking it directly to the probability with which the scenario solution violates the chance constraint (1.2b). Moreover, [9, 12] show that the theory can be extended to the case where sampled constraints are discarded a posteriori, that is after observing the outcomes of the samples. While this increases the complexity of the scenario approach (in terms of data requirement and computation), it can be used to improve the objective value achieved by the scenario solution. In fact, the scenario solution can be shown to converge to the exact solution of (1.2) when the number of discarded constraints are increased, given that some mild technical assumptions hold, cf. [12, Sec. 4.4]

1.3 Novel Contributions

From a practical point of view, the strongest appeal of the scenario approach is the facility of its application and the low computational complexity. It becomes particularly attractive for uncertain optimization problems in higher dimensions, as these occur frequently in fields such as engineering or logistics. In these cases, an uncertain constraint will often not involve all decision variables simultaneously, as allowed by the general case of (1.2b). Instead, multiple uncertain constraints may be present, each of them involving only a subset of the decision variables.

Example 1.3 (Multi-Stage Decision Problems)

An important example are uncertain multi-stage decision problems [5, Cha. 7], [14, Cha. 8] [19, Cha. 13] [21, Cha. 3], which occur in many fields such as production planning, portfolio optimization, or control theory. The basic setting is that some decision (e.g. on production quantities, buy/sell orders, or control inputs) has to be taken repeatedly at a finite number of time steps. Each decision affects the state of the system (e.g. inventory level, portfolio, or state variable) at the subsequent time step. Besides the decision, the state is also subject to uncertain influences (e.g. customer demand, price fluctuations, or dynamic disturbances). If constraints on the state variables are present (e.g. service levels, value at risk, or safety regions), this adds multiple uncertain constraints (one for the state of each time step) to the overall decision problem. Further deterministic constraints may hold for the decision variables, for example. The special structure of such a problem is that a constraint on the state at some time step involves only the decisions made prior to this time step, while the decisions afterwards are not involved.

This paper extends the theory of the scenario approach for problems where a single (or multiple) chance constraint(s) are present that involve only a subset of the decision variables. More precisely, the chance constraint(s) may affect only a certain subspace of the decision space, whose dimension will be called its ‘support rank’. Other constraints, either deterministic or uncertain, cover the directions that are left unconstrained, so that the solution remains bounded.

The main result of this paper is that an uncertain constraint with a lower support rank can only supply a lower number of support constraints [9, 10, 11], and therefore its associated sample size can be reduced. This leads to a subtle shift from the idea of a ‘problem dimension’ in the existing theory to that of a ‘support dimension’ of a particular chance constraint. Moreover, it requires an extension of the existing theory to cope with multiple chance constraints in the uncertain optimization program. Finally, the approach of constraint removal a posteriori is carried over almost analogously to this extended setting.

From a practical point of view, these extensions improve on the merits of the scenario approach for problems that have a structure described above. In particular, the lower sample sizes reduce the computational complexity of the scenario approach and simultaneously improve the objective value of the scenario solution. At the same time, the feasibility guarantees for the scenario solution remain as strong as before. Hence the extensions of this paper, when applicable, offer only advantages over the existing results on the scenario approach.

1.4 Organization of the Paper

Section 2 contains the problem statement. Section 3 introduces some background on its properties, and states the rigorous definitions for the ‘support dimension’ and the ‘support rank’ of a chance constraint. Section 4 contains the main results of this paper, which give the improved sample bounds in the presence of a single (or multiple) chance constraint(s) of limited support rank. Section 5 extends this theory to the sampling-and-discarding procedure, which can be used to improve the objective value of the scenario solution, at the price of larger data requirements and an increased computational complexity. Section 6 presents a brief numerical example that demonstrates the application of the presented theory, as well as its potential benefits when compared to existing results.

2 Problem Formulation

This section introduces the generalized problem formulation with multiple chance constraints, the corresponding scenario program, and some basic terminology.

2.1 Stochastic Program with Multiple Chance Constraints

Consider the following extension of the to an optimization problem with linear objective function and multiple chance constraints ():

(2.1a)
s.t. (2.1b)

where is the chance constraint index in . The remarks for the in Section 1.2 apply analogously; in particular the following key assumption is made.

Assumption 2.1 (Convexity)

The constraint functions of all chance constraints are convex in their first argument for -almost every .

Other than Assumption 2.1, the dependence of the functions on the uncertainty is completely generic.

The use of ‘’ instead of ‘’ in (2.1a) is justified by the fact that the feasible set of a single chance constraint is closed under fairly general assumptions [14, Thm. 2.1]. This implies that the feasible set of the is compact, due to the presence of , and the infimum is indeed attained.

It remains a standing assumption that the -algebra of -measurable sets in is large enough to contain all sets whose probability is measured in this paper, like the ones in (2.1b), cf. [11, p. 4].

In order to avoid technical issues, which are of little relevance for most practical applications, the following is assumed, cf. [11, Ass. 1].

Assumption 2.2 (Existence and Uniqueness)

(a) Problem (2.1) admits at least one feasible point. By the compactness of , this implies that there exists at least one optimal point of (2.1). (b) If there are multiple optimal points of (2.1), a unique one is selected by the help of a tie-break rule (e.g. the lexicographic order on ).

In principle, an approximate solution to the can be obtained by the classic scenario approach. Namely, a can be setup with the same objective function (1.2a) as the , and a chance constraint (1.2b) defined by

(2.2)

Note that is convex in for almost every , since the pointwise maximum of convex functions is convex. Any feasible point of this is also a feasible point of the , and hence an approximate solution to the with confidence is also an approximate solution to the with confidence .

However, this procedure introduces a considerable amount of conservatism, because it requires the scenario solution to simultaneously satisfy all constraints with the highest of all probabilities . Clearly, this conservatism becomes more severe if the number of chance constraints is large and there is a great variation in the values of .

2.2 The Extended Scenario Approach

The extended scenario approach of this paper can be used to compute an approximate solution of the , which is a feasible point of every chance constraint with a given confidence probability of . The key difference from the classic scenario approach is that each chance constraint is sampled separately, and with an individual sample size .

Let the random samples pertaining to constraint be denoted , where , and for brevity also as the collective multi-sample . The collection of all samples is combined in an overall multi-sample , with the total number of samples given by . All of these samples can be considered ‘identical copies’ of the random uncertainty , in the sense that they are themselves random variables and satisfy the following key assumption.

Assumption 2.3 (Independence and Identical Distribution)

The sampling procedure is designed such that the set of all random samples, together with the actual random uncertainty,

form a set of independent and identically distributed (i.i.d.) random variables.

The multi-sample is an element of , the -th product of the uncertainty set , and it is distributed according to , the -th product of the measure . The scenario program for multiple chance constraints () is constructed as follows:

(2.3a)
s.t. (2.3b)

In problem (2.3), the objective function of the is minimized, while forcing to lie inside the constrained sets for all samples substituted into the corresponding constraint . Clearly, the solution to problem (2.3) is itself a random variable, as it depends on the random multi-sample . For this reason, the scenario approach is a randomized method for finding an approximate solution to the .

Of course, the is actually solved for the observations of the random samples, leading to its deterministic instance ():

(2.4a)
s.t. (2.4b)

Note that (2.4) arises from (2.3) by replacing the (random) samples , , with their (deterministic) outcomes , , . Throughout the paper, these outcomes are indicated by a bar, to distinguish them from the corresponding random variables. By Assumption (2.1), constitutes a convex program that can be solved efficiently by a suitable algorithm for convex optimization, cf. [7, 16, 18].

Note that (2.3) remains important for analyzing the (probabilistic) properties of the (random) scenario solution. In fact, the subsequent theory is mainly concerned with showing that, with a very high confidence, the scenario solution is a feasible point of the chance constraints (2.1b), provided that the sample sizes are appropriately selected.

2.3 Randomized Solution and Violation Probability

In order to avoid unnecessary complications, the following technical assumption ensures that there always exists a feasible solution to the , cf. [11, p. 3].

Assumption 2.4 (Feasibility)

(a) For any number of samples , the admits a feasible solution almost surely. (b) For the sake of notational simplicity, any -null set for which (a) may not hold is assumed to be removed from .

Assumption 2.4 can be taken for granted in the majority of practical problems. When it does not hold in a particular case, a generalization of the presented theory accounting for the infeasible case can be developed along the lines of [9].

Hence the existence of a solution to is ensured, and uniqueness holds by Assumption 2.1 and by carry-over of the tie-break rule of Assumption 2.2(b), see [20, Thm. 10.1, 7.1]. Therefore the solution map

(2.5)

is well-defined, returning the unique optimal point of the for a given outcome of the multi-samples . The solution map can also be applied to the , for which it is denoted by . Now represents a random vector of unknown probability distribution, which is also referred to as the scenario solution. In fact, its distribution is a complicated function of the geometry and the parameters of the problem.

Note that there are two levels of randomness present in the analysis. The first is introduced by the random samples in , which affect the choice of the scenario solution. The second is the actual random uncertainty , which determines whether or not the scenario solution is feasible with respect to the chance constraints (2.3b). For this reason, the scenario approach presented here is also called a double-level-of-probability approach [8, Rem. 2.3].

To highlight the two probability levels more clearly, suppose first that the multi-sample has already been observed, so that the scenario solution is fixed. Then for each chance constraint in (2.1b), the a posteriori violation probability is given by

(2.6)

In particular, each has a deterministic, yet generally unknown, value in . If the multi-sample has not yet been observed, the scenario solution is a random vector and so the a priori violation probability

(2.7)

becomes itself a random variable on , with support . Hence the goal is to choose appropriate sample sizes which ensure that for all , with a sufficiently high confidence . Before these results are derived however, some structural properties of scenario programs and technical lemmas ought to be discussed.

3 Structural Properties of the Constraints

In this section, a structural property of a chance constraint is introduced which yields a reduction in the number of samples below the levels given by the existing theory [10, 9, 11]. This property relates to the new concept of the support dimension or, in a form that is more easily checked for many practical instances, the support rank.

3.1 Support Constraints

The concept of a support constraint carries over from the case, cf. [10, Def. 4]. An illustration is given in Figure 3.1.

Definition 3.1 (Support Constraint)

Consider the for some outcome of the multi-sample . (a) For some and , constraint is a support constraint of (2.4) if its removal from the problem entails a change in the optimal solution:

In this case the sample is also said ‘to generate this support constraint.’ (b) For each , the indices of all samples that generate a support constraint of the are included in the set . Moreover, the tuples of all support constraints of the are collected in the support (constraint) set . With some abuse of this notation, .

Definition 3.1(a) can be stated equivalently in terms of the objective function: a sampled constraint is a support constraint if and only if the optimal objective function value (or its preference by the tie-break rule) is strictly larger than when the constraint were removed. To be more precise, Definition 3.1(b), may also account for the set as an additional support constraint. This minor subtlety is tacitly understood in the sequel.

(a) b
(b) b
(c) b
Figure 3.1: Illustration of Definition 3.1 in . The arrow indicates the optimization direction, the bold lines are the support constraints of the respective configuration.

In the stochastic setting of the , whether or not a particular random sample generates a support constraint becomes a random event, which can be associated with a certain probability. Similarly, the support constraint set , and its subsets contributed by the various chance constraints, are naturally random sets.

3.2 Support Dimension

The link between the sample sizes and the corresponding violation probability of the scenario solution depends decisively on the ‘dimensions’ of the problem. The following lower bounds represent a mild technical condition, cf. [9, Thm. 3.3] and [11, Def. 2.3].

Assumption 3.2

The sample sizes satisfy .

In the existing literature, the dimension of the has been characterized by Helly’s dimension, cf. [9, Def. 3.1]. In this paper, there is a subtle shift from the problem dimension to the dimension of chance constraint in the , embodied by its support dimension.

Definition 3.3 (Support Dimension)

(a) Denote by the (random) cardinality of the set . Helly’s dimension is the smallest integer that satisfies

(b) The support dimension of a chance constraint in the is the smallest integer that satisfies

From a basic argument using Helly’s Theorem, the number of support constraints of any (feasible) convex optimization problem in is upper bounded by the dimension of the decision space , cf. [10, Thm. 2]. This result implies that finite integers and matching Definition 3.3 always exist, so that the concepts of ‘Helly’s dimension’ and ‘support dimension’ are indeed well-defined. Moreover, the result provides immediate upper bounds on the support dimension of each chance constraint in (2.3), namely .

It turns out that the support dimension directly relates to the minimum sample size that is required for a given violation level and residual probability . The basic mechanism shall be illustrated by the proposition below, for the simpler case of a single-level of probability problem, cf. [10, Thm. 1].

Proposition 3.4 (Probability Bound)

Consider a particular constraint in the with some fixed sample size , and let be an upper bound for its support dimension . Then the following holds:

(3.1)
Proof.

Consider and let denote the set of support constraints generated by samples from , where stands for generating a support constraint. Note that the event where can be equivalently expressed as generating a support constraint of . Hence condition (3.1) can be reformulated as

(3.2)

To analyze the event , observe that by Assumption 2.3 all samples in are i.i.d., whence all sampled instances of constraint in (2.3b) along with ‘’ are probabilistically identical. In particular, they are all equally likely to become a support constraint of . Hence if the number of support constraints were known, then

Even though is a random variable, by Definition 3.3(b) almost surely, and by assumption . This immediately yields (3.1). ∎

3.3 The Support Rank

In many practical cases, the support dimension of a chance constraint in the is not known exactly. Then it has to be replaced by some upper bound. As argued above, the existing upper bound is given by the dimension of the decision space. However, this bound may not be tight in the case where the constraints satisfy a certain structural property, namely when they have a limited support rank.

Intuitively speaking, the support rank is the dimension of the decision space less the maximal dimension of an (almost surely) unconstrained subspace. The latter is understood as a linear subspace of that cannot be constrained by the sampled instances of constraint , for almost every value of the multi-sample .

Before the support rank is introduced in a rigorous manner, three examples of constraint classes with bounded support rank are described, in order to equip the reader with the necessary intuition behind this concept. They also show that very common constraint classes possess this property, and that in practical problems it can often be spotted easily.

Example 3.5

For each of the following cases, a visual illustration can be found in Figure 3.2.

(a) Single Linear Constraint. Suppose some chance constraint of (2.1b) takes the linear form

(3.3)

where , and is a scalar depending on the uncertainty in a generic way. Note that these constraints in the are unable to constrain any direction in the subspace orthogonal to the span of , , regardless of the outcome of the multi-sample . Hence the support rank of the chance constraint (3.3) is equal to .

(b) Multiple Linear Constraints. As a generalization of case (a), suppose that some chance constraint of (2.1b) is given by

(3.4)

where and represent a matrix and a vector that depend on the uncertainty . Moreover, suppose that the uncertainty enters the matrix in such a way that the dimension of the linear span of its rows , for , satisfies

Note that these constraints in the are unable to constrain any direction in , regardless of the outcome of the multi-sample . Hence the support rank of the chance constraint (3.4) is equal to .

(c) Quadratic Constraint. For a nonlinear example, consider the case where some chance constraint of (2.1b) is given by

(3.5)

where is positive semi-definite with , and , represent a vector and scalar that depend on the uncertainty. Note that these constraints in the are unable to constrain any direction in the null space of the matrix , regardless of the outcome of the multi-sample . Since this null space has dimension , the support rank of the chance constraint (3.5) is equal to .

To introduce the support rank in a rigorous manner, pick a chance constraint of the . For each point and each uncertainty , denote the corresponding level set of by

(3.6)
(a) b
(b) b
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(c) b
Figure 3.2: Illustration of Example 3.5 in . The arrows indicate the dimension of the unconstrained subspace, equal to minus the respective support rank , , or .

Let be the collection of all linear subspaces in . In order to be unconstrained, select only those subspaces that are contained in almost all level sets :

(3.7)

Introduce ‘’ as the partial order on defined by set inclusion; i.e. for any two subspaces , if and only if . Then the following concepts are well-defined, as shown in Proposition 3.7 below.

Definition 3.6 (Unconstrained Subspace, Support Rank)

(a) The unconstrained subspace of chance constraint is the unique maximal element in , in the sense that for all . (b) The support rank of chance constraint equals to minus the dimension of ,

It is a minor technicality in Definition 3.6 that any -null set that adversely influences the dimension of the unconstrained subspace can be removed from ; this is tacitly understood.

Observe that if contains only the trivial subspace, then the support rank is actually equal to Helly’s dimension . On the other hand, if contains more than the trivial subspace, then the support rank becomes strictly less than .

Proposition 3.7 (Well-Definedness of Unconstrained Subspace)

The collection contains a unique maximal element in the set-inclusion sense, i.e. contains all other elements of as subsets.

Proof.

First, note that is always non-empty, because for every and every the level set includes the origin by its definition in (3.6). Therefore contains (at least) the trivial subspace .

Second, since every chain in has an upper bound (namely ), Zorn’s Lemma (or the Axiom of Choice, cf. [6, p. 50]) implies that has at least one maximal element in the ‘’-sense.

Third, in order to prove that the maximal element is unique, suppose that are two maximal elements of . It will be shown that their direct sum , so that would contradict their maximality. According to (3.7), it must be shown that for any fixed values and . To see this, pick

Then apply (3.6) twice to obtain

because and . ∎

3.4 The Support Rank Lemma

The following lemma provides the link between the support rank of a chance constraint and its support dimension.

Lemma 3.8 (Support Rank)

Suppose that a chance constraint has the support rank . Then its support dimension in the is bounded by .

Proof.

Without loss of generality, the proof is given for the first chance constraint . Pick any random multi-sample (less any -null set for which the support rank condition may not hold).

By the assumption, there exists a linear subspace of dimension for which

The orthogonal complement of , , is also a linear subspace of with dimension , and every vector in can be uniquely written as the orthogonal sum of vectors in and , cf. [6, p. 135].

For the sake of a contradiction, suppose that contributes more than support constraints to the resulting , i.e. . For any , let

be the solution obtained if this support constraint is omitted. By Definition 3.1, if a support constraint is omitted from , its solution moves away from , i.e. for all . Denote the collection of all solutions by

so that . Observe that each is feasible with respect to all constraints of the , except for the one generated by , which is necessarily violated according to Definition 3.1.

Since is the orthogonal direct sum of and , for each point in there is a unique orthogonal decomposition of

where . Consider the set

By the hypothesis, contains at least distinct points in the -dimensional subspace . According to Radon’s Theorem [23, p. 151], can be split into two disjoint subsets, and , such that there exists a point in the intersection of their convex hulls:

(3.8)

Split the indices in correspondingly into and , and observe that every satisfies the constraints in :

The last implication follows because and is convex. Similarly, every point satisfies the constraints in :

Combining both statements thus yields

(3.9)

According to (3.8), can be expressed as a convex combination of elements in or . Splitting the points in into and correspondingly and applying the same convex combination yields some

(3.10)

and thereby also some with .

To establish the contradiction two things remain to be verified: first that is feasible with respect to all constraints, and second that it has a lower cost (or a better tie-break value) than . For the first, because all points of lie in and . Moreover, thanks to (3.9),

For the second, pick the set from and that does not contain ; without loss of generality, say this is . By construction, all elements of have a strictly lower objective function value (or at least a better tie-break value) than . By linearity this also holds for all points in , where lies according to (3.10). ∎

Remark 3.9 (Support Rank versus Support Dimension)

While the support rank is a property of chance constraint alone, the support dimension may depend on the overall setup of the . The support dimension constitutes the relevant basis for selecting the sample size . However, it may be difficult to determine for practical problems, as it may depend on the interactions of multiple chance constraints (see Example 3.10 below). The support rank provides an easier-to-handle upper bound to , which can be used in place of for selecting .

Example 3.10 (Upper Bounding of Support Dimension)

To illustrate the statements in Remark 3.9, consider a small example of (2.1) in dimension . Let be the unit cube, with a lexicographic tie-break rule, and two chance constraints . Both constraints affect only the first and second coordinates and , leaving the choice of for the third coordinate. For , the constraints are parallel hyperplanes constraining from below, where the lower bound is given by the first uncertainty :

For , the constraints are V-shaped, with the vertex located at and :

Both uncertainties are uniformly distributed on the interval . The setup is illustrated in Figure 3.3.

In this case, the support dimensions are , and the support ranks are , for the constraints . Notice that for the support rank is strictly greater that its support dimension, due to the presence of constraint . Hence there is some conservatism in the upper bound, although both bounds are better than the existing upper bound by the dimension of the decision space [10, Thm. 2].

Figure 3.3: Illustration of Example 3.10. The plot shows a projection on the -plane for . The unit box is depicted by a dotted line. Two (possible) samples are shown for the linear constraint () and for the V-shaped constraint ().

4 Feasibility of the Scenario Solution

In the first part of this section, it is shown that for a proper choice of the sample sizes the scenario solution is an approximate solution of the (i.e. it is a feasible point of each chance constraint in (2.1b) with a high confidence ). In the second part of this section, an explicit formula for computing the sample sizes for given residual probabilities is provided.

4.1 The Sampling Theorem

Denote by the beta distribution function, cf. [1, p. 26.5.3, 26.5.7]:

(4.1)
Theorem 4.1 (Sampling Theorem)

Consider problem (2.3) under Assumptions 2.1, 2.2, 2.3, 2.4, 3.2. Then

(4.2)

for each chance constraint , whose support rank is .

Proof.

The result is an extension of [11, Thm. 2.4] for the classic scenario approach, which is also used as a basis for this proof.111The authors thank an anonymous reviewer for his/her helpful suggestions on simplifying the proof.

Without loss of generality, consider the first chance constraint ; the result for the other chance constraints follows analogously. Consider the conditional probability

(4.3)

i.e. the probability of drawing such that has a probability of violating ‘’ that is higher than , given fixed values for the other samples .

Clearly, the quantity in (4.3) generally depends on the multi-samples . However, for -almost every value of these multi-samples (4.3) can be bounded by

(4.4)

Indeed, by Assumption 2.1, for -almost every the function defined by

is convex, as it is the point-wise maximum of convex functions. Then all sampled constraints of can be expressed as the deterministic convex constraint ‘’, which can be considered as part of the convex set . Thus for -almost every the problem takes the form of a classic , to which the results of [11] apply. In particular, [11, Thm. 2.4] yields (4.4) for -almost every .

The difference from using the support rank in place of the optimization dimension in [11, Thm. 2.4] is minor. The key fact is that provides an upper bound for the number of support constraints contributed by constraint , according to Lemma 3.8, and hence it can replace in [11, Prop. 2.2] and all subsequent results.

The final result is obtained by deconditioning the probability in (4.3):

based on [22, pp. 183,222], where the third line uses (4.4). ∎

4.2 Explicit Bounds on the Sample Sizes

Formula (4.2) in Theorem 4.1 ensures that with a confidence level of , the violation probability