Preference Elicitation and Robust Optimization with Multi-Attribute Quasi-Concave Choice Functions

05/17/2018 ∙ by William B. Haskell, et al. ∙ University of Southampton National University of Singapore 0

Decision maker's preferences are often captured by some choice functions which are used to rank prospects. In this paper, we consider ambiguity in choice functions over a multi-attribute prospect space. Our main result is a robust preference model where the optimal decision is based on the worst-case choice function from an ambiguity set constructed through preference elicitation with pairwise comparisons of prospects. Differing from existing works in the area, our focus is on quasi-concave choice functions rather than concave functions and this enables us to cover a wide range of utility/risk preference problems including multi-attribute expected utility and S-shaped aspirational risk preferences. The robust choice function is increasing and quasi-concave but not necessarily translation invariant, a key property of monetary risk measures. We propose two approaches based respectively on the support functions and level functions of quasi-concave functions to develop tractable formulations of the maximin preference robust optimization model. The former gives rise to a mixed integer linear programming problem whereas the latter is equivalent to solving a sequence of convex risk minimization problems. To assess the effectiveness of the proposed robust preference optimization model and numerical schemes, we apply them to a security budget allocation problem and report some preliminary results from experiments.

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1 Introduction

Decision making under uncertainty is a universal theme in the stochastic and robust optimization communities. Much of the focus has been on exogenous uncertainties which go beyond the control of the decision maker such as market demand and climate change. In practice, however, there is often significant endogenous uncertainty which arises from ambiguity about the decision maker’s preferences; either because the decision making involves several stakeholders who are unable to reach a consensus, or there is inadequate information for the decision maker to identify a unique utility/risk function which precisely characterizes his preferences. Preference robust optimization (PRO) models are subsequently proposed where the optimal decision is based on the worst preference from an ambiguity set of utility/risk preference functions constructed through available partial information. The robustness is aimed to mitigate the risk arising from ambiguity about the decision maker’s preferences.

Armbruster and Delage [5] study a PRO model for utility maximization problems. Specifically, they model decision maker’s ambiguity about utility preferences by incorporating various possible properties of the utility function such as monotonicity, concavity, and S-shapedness, along with preference elicitation information from pairwise comparisons. An important component their research is to derive tractable formulations of the resulting maximin problem and they manage to do so by exploiting linear envelopes of convex/concave functions. Delage and Li [19] extend this research to risk management problem in finance where the investor’s choice of a risk measure is ambiguous. As in [5], they consider the ambiguity set of risk measures primarily via pairwise elicitation but also featured with important properties such as convexity, coherence, and law invariance, and they develop tractable formulations accordingly.

Hu and Mehrotra [35]

approach the PRO model in a different manner. First, they propose a moment-type approach which allows one to define the ambiguity set for a decision maker’s utility preferences via the certainty equivalent method, pairwise comparisons, upper and lower bounds of the trajectories of the utility functions, and bounds on their derivatives at specified grid points; second, they consider a probabilistic representation of the class of increasing convex utility functions by confining them to a compact interval and scaling them to being bounded by one; third, by constructing a piecewise linear approximation of the trajectories of the utility bounds, they derive a tractable reformulation of the resulting PRO as a linear programming problem. Qualitative convergence analysis is presented to justify the piecewise linear approximation.

Hu and Mehrotra’s approach is closely related to stochastic dominance, a subject which has been intensely studied over the past few decades, see the monographs [46, 55] for a comprehensive treatment of the topic and [20, 21] for the optimization models with stochastic dominance constraints. Indeed, when the preference of a decision maker satisfies certain axioms including completeness, transitivity, continuity and independence, Von Neumann and Morgenstern’s expected utility theory ([63]) guarantees that any set of preferences that the decision maker may have among uncertain/risky prospects can be characterized by an expected utility measure. The issue that we are looking at here is that the decision maker does not necessarily have complete information about his preferences, and prospects associated with decisions do not necessarily have definitive stochastic dominance relationships - which means that the existing models based on stochastic dominance are not applicable.

In a more recent development, Haskell et al. [32] consider a stochastic optimization problem where the decision maker faces both exogenous uncertainty and endogenous uncertainty associated with decision maker’s risk attitude. They propose a PRO model where the ambiguity is constructed in the product space of exogenous uncertainty and utility functions. By using Lagrangian duality, they derive an exact tractable reformulation for some special cases and a tractable relaxation in the general setting. Delage et al. [18]

propose a robust shortfall risk measure model to tackle the case where investors are ambiguous about their utility loss functions. They study viable ways to identify the ambiguity set of loss functions via pairwise comparison for utility risk measures with respective features such as coherence, convexity, and boundedness, and they derive tractable linear program reformulation for the resulting optimization problems with robust shortfall risk constraints.

In this paper, we take on this stream of research but with a different focus in terms of both modeling and tractable reformulations. Specifically, we consider a class of so-called choice functions applicable to multi-attribute decision making which are monotonically increasing along some specified direction, quasi-concave, but not necessarily convex/concave or translation invariant. We tackle the subsequent PRO model via “hockey-stick” type support functions and level functions. Monotonicity ensures that the decision maker universally prefers more to less. Quasi-concavity is a general form of risk aversion, the axiom of quasi-concavity is further supported by the study in [12] on aspirational preferences. Moreover, by dropping translation invariance) which is an key axiomatic property of convex risk measures), we allow our PRO model to cover a wider range of problems where translation invariance may not hold [12].

Another important departure from existing research is that our PRO model is applicable to multi-attribute decision making problems. These problems are ubiquitous in practical applications but the existing PRO models mainly emphasize the single attribute setting. For instance, in management research of healthcare, it is typical to use several metrics rather than just one to measure the quality of life [58, 24, 56]. Similar problems can be found in network management problems [7, 15], scheduling, [41, 69], design [59, 22] and portfolio optimization [27]. Indeed, over the past few decade, there have been significant research on multi-attribute expected utility [64, 26, 45, 60, 61, 62] and multi-attribute risk management [36, 14, 31, 29]. In a more recent development, research on robust multi-attribute choice models has also emerged, see [39, 23, 49] for instance. In particular, [49] considers optimization with a general class of scalarization functions (in particular, the class of min-biaffine functions), where the weights of the scalarization lie in a convex ambiguity set.

We summarize the main contributions of our present paper as follows.

  • Multi-attribute quasi-concave PRO model. We propose a robust choice model for preference ambiguity where the underlying choice function is monotonic and quasi-concave. By replacing concavity with quasi-concavity and dropping translation invariance, we extend the existing PRO model so that it is easier to incorporate preference elicitation information. Moreover, the new model framework covers a number of well-known preference models (such as expected utility and aspirational preferences) and can be applied to a wider range of PRO problems where the decision maker’s choice function is merely increasing and quasi-concave. Of course, it also poses new challenges to tractable reformulation which so far have depended on convexity [5, 33, 19]. Our model’s support for multi-objective stochastic decision making problems makes the model applicable to an even broader class of problems.

  • New forms of tractable formulations. We propose two schemes for tractable reformulation of the proposed new PRO model. One is based on the support functions of quasi-concave functions and the other exploits approximation of quasi-concave function by a sequence of convex level functions. While both approaches are well known in the literature of generalized convex optimization, they are applied to PRO here for the first time. The support function approach takes the PRO model to a mixed-integer linear program as opposed to a linear program as in earlier work. The level set representation is a generalization of the representation results in [13, 12] to the case of preference ambiguity. We are able to explicitly derive the connection between these two approaches by using the special form of piecewise linear support functions for quasi-concave functions. This framework, based on representing any monotonic diversification favoring choice function in terms of a family of risk functions, leads to a unifying framework for representing multi-attribute choice functions. Moreover, this framework naturally converts a decision-making problem with a quasi-concave choice function into a sequence of convex optimization problems, yielding a viable computational recipe. This development is related to the methods in [13, 12] and extends these methods to the multi-attribute setting. The level set representation further builds on the managerial insights from [13, 12]. It reveals that multi-attribute choice functions can in general be understood in terms of a family of multi-attribute risk functions and the decision maker’s desired satiation levels.

  • Level function method. In the case where the attributes depend nonlinearly on the decision variables, we propose an algorithm for solving the PRO by using the level function method from [68]. The algorithm is an iterative regime where at each iterate we solve a mixed-integer linear program based on the support function representation and identification of a level function, which is closely related to the level set representation. To examine the performance of the model and numerical schemes, we apply them to a homeland security problem considered by Hu and Mehrotra [34]. Our numerical experiments for this problem illustrate how our PRO model captures diversification favoring behavior, and also how it depends on the elicited comparison data set.

The rest of the paper is organized as follows. In Section 2, we review some preliminary materials related to choice functions. In Section 3, we formally describe the robust choice function model including the definition of the class of choice functions to be considered, specification of the ambiguity set, and characterization of the robust choice function and the corresponding maximin optimization problem. Section 4 details the support function approach for tractable reformulation of the robust choice model and its underlying theory. Next in Section 5, we discuss an alternative level set representation for quasi-concave choice functions. This development leads to a general representation formula for quasi-concave choice functions. Here, we also discuss the role of “targets” which feature prominently in the decision analysis literature. Section 6 builds on our PRO model and explains how to solve optimization problems in the presence of preference ambiguity, and Section 7 applies this methodology to a budget allocation problem for homeland security. The paper concludes in Section 8 with a discussion of potential impacts and future research directions.

2 Preliminaries

This section presents some preliminary materials on choice functions. We begin with a set of states of nature endowed with a algebra . Let be an admissible space of measurable mappings , equipped with the supremum norm topology. We generally treat as a space of multi-attribute prospects with characteristics, although the case is also covered. The inequality for is understood to mean component-wise for all . We adopt the convention that prospects in represent rewards/gains so that larger values of all attributes are preferred to smaller values.

Let be the extended-valued real line. A choice function is a mapping that gives numerical values to prospects in to evaluate their fitness (e.g. see [13]). When , is said to be weakly preferred to and when , is said to be strongly preferred to . The following definition specifies the key properties of choice functions that appear frequently in the literature (see [54, 13] for example).

Definition 2.1 (Properties of the choice function).

(i) (Upper semi-continuity) For all , .

(ii) (Monotonicity) For all , implies .

(iii) (Quasi-concavity) For all , for all .

(iv) (Completeness) For all , either or .

Upper semi-continuity is a common technical condition (see [8, 12, 65]). Monotonicity means that the decision maker always prefers more reward to less - this property is universally accepted. Quasi-concavity means that diversification does not decrease reward, where the convex combination is understood as a mixture of the prospects and , see [12, 13]. Properties (i) - (iii) appear in much of the decision theory literature (e.g. [12]). Since the choice function is real-valued, it is automatically transitive: for all if and , then . Property (iv) and transitivity ensure that the choice functions we consider are “rational”.

We now give some examples of multi-attribute choice functions to motivate our discussion and to point out related work on multi-attribute prospects. Here and throughout, the Euclidean inner product is denoted by .

Example 2.2.

Let for be univariate utility functions.

(i) The assumption of mutual utility independence (see [3] for example) gives rise to additive utility functions for

. The expected utility of a random vector

is then .

(ii) In [37], an alternative independent utility aggregation model is proposed with

where and The expected utility is then

(iii) For a general utility function , the expected utility is a choice function.

(iv) Let be a multivariate copula (i.e.

is the joint cumulative distribution function of an

dimensional random vector on the unit cube

with uniformly distributed marginals), then

is a choice function (see [2]).

(v) The conditional value-at-risk (CVaR) of a univariate random variable

at level is

A multivariate version of the conditional value-at-risk (CVaR) is developed in [48] based on linear scalarization. Given a vector of weights , we may consider the choice function on .

(vi) More generally, as in [49], we may take any univariate risk measure (such as a mean-deviation risk measure, see [54]) and then consider the choice function where is a scalarization function. In [49], the authors focus on the computationally tractable class of “min-biaffine” scalarization functions .

3 Robust preferences model

Now we come to the main problem under consideration in this paper. To begin, we introduce the set of all upper semi-continuous, monotonic, and quasi-concave choice functions

This set characterizes our decision makers of interest. Since concave functions are quasi-concave, this class of functions consists of all continuous increasing concave utility functions in the literature.

In practice it is difficult to elicit a precise functional form for . This difficulty is exacerbated in the multi-attribute setting. First, when multi-attribute prospects are in play, it is not obvious how to characterize the marginal dependencies of the variety of attributes, i.e., it is not always clear how much an increase in the value of one asset should depend on the levels of the other assets. Second, it is hard to specify a choice function in group decision making where the group must come to a consensus. Third, we may have only a few observations of the decision maker’s behavior which makes it impossible to precisely specify preferences.

To circumvent these difficulties, we design a robust choice function. To this end, we will first need to construct a preference ambiguity set which contains a range of possible choice functions. Then, given this preference ambiguity set , we will set up a framework which chooses a “robust choice function” as:

(1)

In the upcoming definition, we generalize formulation (1) by taking a “benchmark” prospect . Benchmark prospects have a long history in the field of optimization with stochastic constraints, see for instance [21, 48, 5, 49].

Definition 3.1 (Robust choice function).

Let and be given, then defined via

(2)

is the robust choice function corresponding to and .

When the benchmark is a constant act and all are normalized to have the same value at , we recover formulation (1) from (2). The robust formulation is based on the minimal excess value of over the value of the benchmark prospect for the whole set of choice functions specified in . This kind of conservatism is to be used in decision making for countering risks arising from ambiguity about the true preferences. It is also very much in line with the philosophy of robust optimization. In both [5] and [19], the “worst-case utility function” and “worst-case risk measure” are considered in the same manner. This kind of framework is particularly relevant in the context of group decision making whereby the least favorable utility function from a member of the group is to be used for the holistic decision making process.

Remark 3.2.

In our upcoming development, one may omit the benchmark and just consider the function with only minor modification. We include the benchmark to stay consistent with [5] and the wider literature on stochastic dominance constrained optimization. The presence or absence of the benchmark does not materially affect our main development.

The following proposition shows that the robust choice function itself belongs to whenever .

Proposition 3.3.

For any and , is upper semi-continuous, increasing, and quasi-concave.

Proof.

Upper semi-continuity: Upper semi-continuity of a set of functions is preserved by taking the point-wise infimum of the collection.

Monotonicity: Monotonicity follows by

since for all whenever .

Quasi-concavity: Quasi-concavity follows since

for any with , where the inequality follows by quasi-concavity of all , and the third equality follows by interchanging the order of minimization. ∎

We now turn to discuss specification of the ambiguity set . This set will have the following characteristics:

  • Preference elicitation: For a sequence of pairs of prospects , where is a finite index set, the decision maker prefers to for all . In this case, all admissible choice functions in that are consistent with the decision maker’s observed behaviors must satisfy for all . This form of preference elicitation also appears in [5, 19].

  • Normalization: the decision maker’s choice function satisfies .

  • Lipschitz continuity: the decision maker’s choice function is Lipschitz continuous. Lipschitz continuity ensures that the choice function does not vary too rapidly. Additionally, this technical condition is necessary to apply a key representation result for quasi-concave functions that appears in the next section. Since for all and any , we may specify the Lipschitz constant of arbitrarily.

Our resulting specific ambiguity set is then

We impose the Lipschitz condition since otherwise the set is a cone, in which case may not be finite-valued. In the next section we will develop a computational recipe for evaluating .

We conclude this section by introducing our robust choice model. Let be a set of available decisions and let be a random-variable-valued mapping with realizations denoted for all . The mapping captures the randomness inherent in the underlying decision-making problem. In general, we are interested in solving

(3)

We make the following two key convexity assumptions on the problem data and .

Assumption 3.4.

(i) is closed and convex.

(ii) is concave in the sense that is concave in for almost all .

Assumption 3.4 ensures that our upcoming optimization problems are convex, and the next proposition reveals the subsequent key structural results of Problem (3).

Proposition 3.5.

Problem (3) is a quasi-concave maximization problem.

Proof.

Let and . For any we have

where the first inequality follows by monotonicity of and concavity of , and the second inequality follows by quasi-concavity of . The conclusion then follows by the previous part, the fact that the infimum of quasi-concave functions is quasi-concave, and the fact that the feasible region of Problem (3) is convex. ∎

We will return to Problem (3) later in Section 6 after we carefully examine the function .

4 Support function representation

In this section we turn to the primary issue of numerical evaluation of the robust choice function . At first glance, it is evident that cannot be evaluated with convex optimization because quasi-concavity is not preserved under convex combination (and calls a minimization problem over a subset of quasi-concave functions). In contrast, the robust choice functions in [5] and [19] are amenable to convex optimization because concave utility functions and convex risk measures are preserved under convex combination. Despite this new difficulty, we can build on the support function technique used in [5, 19] and augment it for our present setting in .

We begin with some preliminary definitions and facts associated with support functions. For emphasis, in the following definition and throughout we consider upper support functions which dominate a target function from above rather than below (as in the convex and quasi-convex cases). We remind the reader that denotes the Euclidean inner product.

Definition 4.1.

Let . Recall that is said to be majorized by a function if

and is an upper support function of at if . Here and later on, denotes the domain of . A vector is called a subgradient of at if

We denote the set of subgradients of at by and call the latter subdifferential. A vector is an upper subgradient of at if

We denote the subdifferential of at by .

When is concave and subdifferentiable at , i.e., , the linear function

is a support function of at for any . The following theorem gives rise to a characterization of concave functions by their support functions.

Theorem 4.2.

Let . The following assertions hold.

  • is a concave function if and only if there exists an index set such that

    (4)

    where is possibly infinite and for all .

  • For any finite set and values , defined by

    is concave. Moreover, for all concave functions with , .

Proof.

Part (i). These results are well known, see for instance [11, Section 3.2] and [53].

Part (ii). It is immediate that as defined is concave, as it is of the form (4). Note that the hypograph of is the convex hull of . The convex hull of a set of points is by definition the smallest convex set containing these points. See also [11, Section 6.5.5]. ∎

Theorem 4.2 says that one can recover concave functions by taking the infimum of their support functions. Moreover, it shows that support functions can be used for the construction of the “lowest” concave function that dominates a fixed set of values. The results provide the basis for tractable formulations of PRO models in [5, 33, 19]. For further details on other applications of this result, we refer the reader to [11, Section 6.5.5]

for a discussion of interpolation with convex functions.

When is quasi-concave and upper subdifferentiable at , the piecewise linear function

is a support function of at for any . Note that these functions are the maximum of two linear functions and so open upwards. We informally refer to such as “hockey stick” functions in recognition of this shape. Note that these functions are quasi-concave themselves.

For any and , the function defined by

is quasi-concave since its upper level sets are convex. In particular, it is easy to verify that for ,

and for ,

which is convex (it is a half-space) since it is the upper level set of an affine function.

The following result is the analog of Theorem 4.2 for quasi-concave functions, it characterizes quasi-concave functions via this class of “hockey stick” support functions. The result will provide a theoretical foundation for tractable reformulation of our quasi-concave robust choice function. We note that the first part of the following theorem requires the stronger assumption that our function of interest is Lipschitz continuous.

Theorem 4.3.

Let . The following assertions hold.

  • Suppose that is quasi-concave and Lipschitz continuous. Then

    (5)

    where is possibly infinite and for , .

  • If has a representation (5), then it is quasi-concave.

  • For any finite set and values , defined by

    s.t.

    is quasi-concave. Furthermore, the graph of is the quasi-concave envelope for the set of points .

Proof.

Part (i). By [51, Theorem 2.3], any Lipschitz continuous quasi-concave function is upper subdifferentiable on its domain. Thus for any under the assumption that is L-Lipschitz, and for any , . Moreover, for any and , is supported by at . By taking the infimum of all support functions defined as such, we have

Furthermore, for any ,

since .

Part (ii). For any we have

each is convex by quasi-concavity of for all , and the intersection of convex sets is convex.

Part (iii). The quasi-concavity of follows from the fact that each function is quasi-concave and the infimum of such functions is also quasi-concave. In what follows, we prove the second part of the statement. For any , each with belongs to because by definition . Since is quasi-concave and has convex upper level sets, we must then have

(6)

where “conv” denotes the convex hull of a set.

For , we may choose appropriate parameters , and such that for all and . To construct such a hockey-stick function, first let be the projection of onto the convex set . Next, by virtue of the separation theorem in convex analysis, there exists such that and for all . Let and . Then

By the definition of , the inequality above implies that . On the other hand, it is easy to verify that . Thus, we arrive at , which enables us to deduce that and subsequently

(7)

A combination of (6) and (7) yields

Since the convex hull is the smallest convex set containing a set of points, it must be that

for any other quasi-concave with for all . This same reasoning holds for all , so for all and . ∎

We note that we may use any norm (not necessarily the Euclidean norm) to enforce Lipschitz continuity in part (i) of Theorem 4.3 since all norms on are equivalent. In parallel to Theorem 4.2, Theorem 4.3 gives conditions where a quasi-concave function can be recovered by taking the infimum of its support functions (which are hockey stick functions in this case). Moreover, Theorems 4.2 and 4.3 give conditions for constructing the “lowest” quasi-concave function that contains a fixed set of values.

4.1 Reformulation as a mixed-integer linear program

Theorem 4.3 gives an explicit form for the “worst-case” quasi-concave function that dominates a set of values over a finite set . In fact, this is exactly what we need to derive a tractable reformulation of . The remaining challenge is to put the correct conditions on the values , which will take the form of an optimization problem. To this effect, for the remainder of this section we introduce the major assumption that the underlying sample space is finite.

Assumption 4.4.

The sample space is finite.

Assumption 4.4 also appears in [5, 33, 19] where it is used for obtaining tractable optimization formulations. However, in the case when is continuous, it is possible to develop a discrete approximation (see [18, Section 5] and [33]).

Under Assumption 4.4, we adopt the convention that a prospect may be identified with the vector of its realizations

This convention first appeared in [19] and depends on a finite sample space . In this way, there is a one-to-one correspondence between elements of and . We now define

to be the union of all the prospects used in the definition of (including the constant prospect which is used in the normalization condition) along with the benchmark .

To proceed on, we let

denote the set of choice functions that take the values on the set . We now evaluate the worst-case choice function with the following procedure:

  1. Set the values of on . We only consider these values at first because they give sufficient information to construct on the rest of , as we will show.

    1. The values must satisfy the majorization characterization for quasi-concave functions (a quasi-concave function is majorized by its hockey stick support function at every point on its graph). This is equivalent to determining if .

    2. The values must satisfy the preference elicitation condition given in the definition of .

    3. The values must satisfy the Lipschitz continuity condition given in the definition of .

  2. Once the values of are fixed on satisfying the above conditions, interpolate using hockey stick support functions to determine the value at any .

This procedure results in the following optimization problem:

(8)
s.t. (9)
(10)
(11)
(12)
(13)
(14)

Intuitively, constraint (9) is the majorization characterization for the values for all ; constraint (10) is the normalization constraint; constraint (11) corresponds to the preference elicitation requirement in the definition of ; constraint (12) requires the support functions used to characterize to be increasing and Lipschitz continuous; constraint (13) requires the support function used to determine the value of at to majorize ; and finally constraint (14) requires the support function used to determine the value of at to be increasing and Lipschitz continuous.

Problem (8) - (14) has several features in common with [5, 19]. In particular, all of these formulations have a support function characterization that ensures convexity/concavity/quasi-concavity, and all of these formulations have constraints corresponding to preference elicitation. The main difference is that the formulations in [5, 19] are based on linear functions while our formulation is built on hockey stick functions.

Remark 4.5.

The convention avoids the requirement that the convex hull of contain the support of in Problem (8) - (14) as in [5, 18].

The next theorem formally verifies the correctness of this formulation.

Theorem 4.6.

Suppose Assumption 4.4 holds. Given , the optimal value of Problem (8) - (14) is equal to .

Proof.

To begin, we may partition the set of choice functions by their values on the finite set . We have the equivalence

where we use , and denotes the cardinality of . This condition is simply saying that can be understood as either minimizing over directly, or first fixing the relevant values on and then minimizing over functions in that coincide with those values.

Next define the sets

and

Then, using the fact that , we have

s.t.

Constraint is just the preference elicitation condition (11) and constraint is just the Lipschitz continuity requirement (12). Both of these conditions only constrain the values of on via . Constraint states that there must exist a function in that takes the values on . This requirement is enforced by: (i) constraint (9), the majorization characterization of the quasi-concavity of ; and (ii) constraint (12), the requirement that the support functions that majorize on be increasing and Lipschitz continuous.

So, it remains to evaluate for fixed , which is explicitly

since the value is fixed for by construction of . Over all increasing quasi-concave functions in , the one minimizing attains the value

which is captured by constraint (13) that requires the support function to majorize . ∎

Problem (8) - (14) is finite-dimensional, but it is not a linear programming problem (or even a convex optimization problem) due to constraints (9) and (13) which require a convex function to be greater than a linear term. However, it is possible to transform Problem (8) - (14) into a mixed-integer linear programming problem (MILP) using standard techniques. We obtain the following MILP where we use a new constant :

(15)
s.t. (16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)

In explanation, constraints (16) and (17) replace the term that appears in the objective of Problem (8) - (14) with linear terms via the epigraphical transformation. Constraints (18) and (19) along with the binary constraints on replace constraints (9) with disjunctive constraints; likewise, constraints (23) and (24) along with the binary constraints on replace constraints (13) with disjunctive constraints.

Remark 4.7.

In terms of computation, Problem (15) - (25) can be effectively solved by Bender’s decomposition as proposed in [16, 52, 57]. In [17], the authors explain that the linear programming relaxation of the MILP is typically a poor approximation due to the big- coefficients. In fact, the binary solutions of LP relaxations are only marginally affected by the addition of continuous variables and the associated constraints. Different choices of will affect the branching process in the algorithm, as well as the constraints in the LP relaxation. In recognition of this difficulty, these authors introduce “Combinatorial Bender’s cuts” that can generate more effective cuts for the master problem and that also avoid the difficulty of choosing the constant . Alternatively, based on [38], we could use a convex hull reformulation for each disjunction in Problem (15) - (25). This reformulation will always give a tighter bound than the big- formulation, at the expense of a large number of variables and constraints. Both of these approaches converge to the global optimum after finitely many iterations, and can improve upon basic methods for solving Problem (15) - (25).

4.2 The worst-case choice function

We conclude this section by giving further details on the explicit form of the worst-case choice function. In [5], the worst-case utility function is shown to be piecewise linear concave by using a support function argument. This is possible because the worst-case utility function in question is a mapping from to . Our present setting is more complicated because: (i) we deal with multi-attribute prospects and (ii) we are concerned with quasi-concave functions. Yet, as we will see shortly, our worst-case choice function can also be constructed explicitly. The Delaunay triangulation, defined next, is the key to this approach.

Definition 4.8.

(i) A simplex is a polytope in such that is the convex hull of affinely independent points.

(ii) A simplicial complex is a finite collection of simplices such that: , is a simplex; and imply ; and for any , either or .

(iii) A triangulation of a finite set is a simplicial complex whose vertices belong to and whose union is the convex hull of .

(iv) A circumsphere (circumscribed sphere) of a simplex is a sphere that contains all of the vertices of .

(v) A Delaunay triangulation of a finite set is a triangulation such that no point in is inside the circumsphere of any simplex in .

We use the Delaunay triangulation of in our construction of the worst-case choice function. The next theorem shows that the worst-case choice function (without Lipschitz continuity) is piecewise constant, and follows directly from Theorem 4.3(iii).

Theorem 4.9.

Choose such that , then is given by

Next we consider the case where Lipschitz continuity is enforced. This result also follows directly from Theorem 4.3(iii), only now the worst-case choice function is piecewise linear.

Theorem 4.10.

Choose such that , then is given by

where for all and , and .

5 Level set representation

The previous section emphasized support functions as a computational tool which leads to an MILP formulation for the robust choice function . In this section, we take an alternative approach and focus on the so-called level set representation for quasi-concave choice functions. This result is important because it generalizes beyond worst-case choice functions and offers a common framework for representation of multi-attribute choice functions. Furthermore, this approach has not yet been seen in the work on robust preference utility/risk models.

5.1 Risk measures

Risk measures are the core ingredient for our level set representation. To begin, we formally define risk measures for the multi-attribute setting, where we are particularly concerned with convex risk measures which play a major role in risk-aware optimization (see [6, 54]).

Definition 5.1.

A function is a convex risk measure if it satisfies

(i) Monotonicity: If then .

(ii) Normalization: .

(iii) Convexity: for any and , .

Since we treat prospects in as gains/rewards, our definition of monotonicity above is the opposite of the typical definition of monotonicity for losses. Convexity of risk measures is extremely important for our considerations because of the prominent role of convexity in optimization. Note that we do not yet stipulate a property of translation invariance for the multi-attribute setting, we give further commentary on this issue later.

Any risk measure induces a set of “acceptable” prospects in the sense that a prospect is acceptable if , i.e. it has nonpositive risk according to . We formalize this idea in the following definition.

Definition 5.2.

Let be a risk measure, the set is the acceptance set associated with .

Note that is a convex set in whenever is a convex risk measure. Conversely, given an acceptance set we can specify a risk measure for some with . We interpret as the vector-valued amount that must be added to to make acceptable to the decision maker.

For a choice function which evaluates the fitness of prospects in , we seek a family of convex risk functions such that:

(26)

Relationship (26) means that the upper level sets of the choice function can be characterized by the acceptance sets of a family of convex risk functions. The acceptance sets are closed and convex since is assumed to be upper-semicontinuous and quasi-concave. From a practical point of view, if a decision maker with choice function selects satisfaction level , then the prospect of exceeding is equivalent to the prospect of the risk being less than or equal to zero under measure . This perspective is related to the notion of satisficing measures developed in [13]. We interpret (26) to mean that the set of satisfiable prospects can be characterized by the acceptance sets of a sequence of risk measures.

In the case of relationship (26), we have the representation

(27)

This equivalence is established in Proposition A.3 in the Appendix. Representation (27) is valuable both from a theoretical perspective and a computational one. Theoretically, it reveals the connection between quasi-concave choice functions and convex risk measures. Computationally, it allows us to evaluate quasi-concave choice functions with a sequence of convex feasibility problems and a bisection algorithm.

Remark 5.3.

A related form of representation (27) is considered for univariate prospects in [12]. As an illustrative example, [12, Example 5] considers the case where the are given by CVaR. Furthermore, the univariate form of Representation (27) can be viewed as a generalization of the shortfall risk measure from [28, Section 4.3], [4], [66, Section 3], and [30]. In this case, for a convex loss function we may take for all .

In representation (27), we may also take to be any support function of at . We now consider the family more carefully, and we require it to satisfy the following assumptions.

Assumption 5.4.

is a family of risk measures which satisfies the following:

  • for each fixed , is monotonically decreasing (non-increasing) on ;

  • for each fixed , is convex on ;

  • for each fixed ,