In this section, we first introduce the notion of a multi-objective online problem and, secondly, define the concept of competitive analysis for multi-objective online problems.
3.2.1 Multi-Objective Online Problems
In the following, we define the concept of competitive analysis for multi-objective online problems with respect tominimization problems. If not mentioned otherwise, the defini-tion for the corresponding maximizadefini-tion problem is analogous. First of all, we define a multi-objective optimization problem P as a triple (I,X, f), consisting of a set of inputs I, a set of feasible outputs (or solutions) X(I) associated with every input I ∈ I, and the objective functionf given asf :I × X →Rn+ where, forx∈ X(I),f(I,x)represents the objective value of the solutionx with respect to inputI ∈ I.
Given inputI ∈ I, analgorithm alg for a multi-objective optimization problemP computes a feasible solution alg[I]∈ X(I). The objective associated with this feasible output is denoted by alg(I) = f(I,alg[I]). According to (Ehrgott, 2005, p. 24), a feasible solution ˆx ∈ X(I) is called efficient if there is no other x ∈ X(I) such that f(I, x)f(I,x), whereˆ denotes a component-wise order, i.e., for x, y∈Rn,xy:⇔ xi≤yi, for i= 1, . . . , n, andx6=y. Anoptimal algorithm optforP is such that, for all inputs I ∈ I,opt[I]is the set of efficient solutions toP, i.e.,
opt[I] ={x∈ X(I)|xis an efficient solution to P}. The objective associated with a solutionx∈opt[I]is denoted byopt(x).
The definition of a multi-objective online problem is now given analogously to the definition of a single-objective online optimization problem given in (Borodin and El-Yaniv, 1998, p. 2). Accordingly, multi-objective online problems are multi-objective optimization problems in which the input is revealed bit by bit and an output must be produced in an online manner, i.e., after each new bit of input a decision affecting the output must be made.
3.2.2 The Competitive Ratio and Competitiveness
The study of online problems is concerned with assessing the quality of corresponding online algorithms and, ultimately, the question of which is the best algorithm. We carry this leading question forward to multi-objective online problems. In the following, we list conditions that are supposed to be met by an appropriate measure for the quality of multi-objective online algorithms:
Condition 1 (worst case model): Just as in the case of competitive analysis for sin-gle-objective algorithms (cf. (Fiat and Woeginger, 1998, p.4)), we aim for a worst case model for multi-objective competitive analysis that holds for any input distri-bution in order to avoid the problems of probabilistic models.
Condition 2 (worst case ratio): Furthermore, a standard worst case analysis of mul-ti-objective online algorithms leads to the same pitfall as in the single-objective case (cf. (Fiat and Woeginger, 1998, p.3)): Due to the incomplete knowledge of the online algorithm, it is often possible to ensure that each decision made by an online algorithm is the worst possible decision with respect to all components. For
example, consider the multi-objective time series search problem (see Section 3.3):
if a sequence consisting only of the minimal price vector is revealed, the online player always ends up with the minimal price vector regardless of his strategy.
Therefore, following the underlying idea of competitive analysis, it is desirable to consider the ratio of the algorithm’s performance and the optimal performance in every component on the same problem instance.
Condition 3 (independence from efficient solutions): Normally, the solution to a multi-objective optimization problem is given by a set of efficient solutions (see, for example, (Ehrgott, 2005)). However, due to the online nature of our approach and the corresponding urge to obtain an autonomous algorithm, we assume a multi-objective online algorithm to compute a single solution instead of a set of solutions.
The competitive ratio should, nevertheless, be independent of a particular solution chosen from the set of efficient solutions of the offline problem.
Condition 4 (total order): In order to compare different multi-objective online algo-rithms, a total order on the competitive ratio of multi-objective online algorithms is necessary.
These requirements lead us to the following definition ofc-competitiveness for multi-objective online algorithms:
Definition 3.2.1. A multi-objective online algorithm alg is c-competitive if, for all finite input sequences I, there exists an efficient solutionx∈opt[I]such that
alg(I)i≤ci·opt(x)i+αi, for i= 1, . . . , n , where c= c1, . . . , cn|
andα∈Rn is a constant vector independent ofI.
Note thatcis a vector instead of a scalar as in the classic definition of competitiveness for single-objective online algorithms. A multi-objective online algorithm which accom-plishes this postulation even for all efficient solutions is calledstrongly c-competitive:
Definition 3.2.2. A multi-objective online algorithm alg is stronglyc-competitive if, for all finite input sequences I and all efficient solutions x∈opt[I],
alg(I)i≤ci·opt(x)i+αi, for i= 1, . . . , n , where c= c1, . . . , cn|
andα∈Rn is a constant vector independent ofI.
Applying these definitions to objective problems results in the classical single-objective competitive ratio for both Definition 3.2.1 and Definition 3.2.2. Obviously, every stronglyc-competitive multi-objective online algorithm is also c-competitive. For maximization problems, the inequalities in Definitions 3.2.1 and 3.2.2 are replaced by alg(I)i ≥1/ci·opt(x)i+αi.
The definition of competitiveness for multi-objective online algorithms is a worst case ratio due to the consideration of all finite input sequences as required by Conditions 1
and 2. Furthermore, the definition takes the set of all efficient offline solutions into account and hence does not rely on a particular efficient solution, as demanded by Con-dition 3. In order to achieve a comparable competitive ratio of multi-objective online algorithms as demanded by Condition 4, a total order on the competitiveness of an online algorithm is necessary. This gives rise to the following definition of the competitive ratio for multi-objective online algorithms:
Definition 3.2.3. Let f :Rn → R+. The infimum over the set of all values f(c) such that alg is (strongly) c-competitive is called the (strong) competitive ratio with respect to f of alg and is denoted by (Rfs(alg))Rf(alg).
The choice of the function f grants a certain degree of freedom that is left to the analyst of the online algorithm (in the style of the decision maker in the field of multi-objective optimization). However,f has to be chosen such thatf(c)≤f(ˆc)if ci≤ˆci for i= 1, . . . , nin order to guarantee a reasonable setting.
In this work, we consider three intuitive choices for the function f. First of all, consider f1 given as f1(c) := maxi=1,...,nci. By this choice, the competitive ratio is guaranteed for each component of the objective function. We label this choice as worst-component competitive ratio. Further, we consider f given by f2(c) := n1 Pn
i=1ci and f3(c) := pQn n
i=1ci. In these cases, the arithmetic and geometric mean value of the com-ponents’ competitive ratios is taken, which is why these choices are labeled as arithmetic-and geometric-mean-component competitive ratio.
For randomized multi-objective online algorithms, the definition of the (strong) com-petitive ratio is given in the same way. Letalg be a randomized multi-objective online algorithm. An oblivious adversary must choose a finite inputI in advance, based on the knowledge of the probability distribution(s)alg uses.
Definition 3.2.4. A randomized multi-objective online algorithm alg is c-competitive against an oblivious adversaryif, for every input I chosen as described above, there exists an efficient solution x∈opt[I] such that
E[alg(I)i]≤ci·opt(x)i+αi, for i= 1, . . . , n, where c= c1, . . . , cn|
and α∈Rn is a constant independent of I.
Definition 3.2.5. A randomized multi-objective online algorithm algis strongly c-com-petitive against an oblivious adversaryif, for every inputI chosen as described above and all efficient solution x∈opt[I],
E[alg(I)i]≤ci·opt(x)i+αi, for i= 1, . . . , n, where c= c1, . . . , cn|
and α∈Rn is a constant independent of I.
Note that E[alg(I)i] is the expected value of the i-th component of alg with re-spect to its randomized decisions. Again, applying these definitions to single-objective problems results in the classical single-objective competitive ratio and, for maximization
problems, the inequalities in Definitions 3.2.4 and 3.2.5 are replaced by E[alg(I)] ≥
1/ci·opt(x) +α.
The definition of the competitive ratio for randomized multi-objective online algo-rithms is given accordingly:
Definition 3.2.6. Let f : Rn → R+. The infimum over the set of all values f(c) such that alg is (strongly) c-competitive against an oblivious adversary is called alg’s (strong) competitive ratio with respect tof against an oblivious adversaryand is denoted by (Rfs(alg))Rf(alg).
In online optimization, two further adversary models for randomized algorithms are known, namely the adaptive-online adversary and the adaptive-offline adversary (see Section 1.1). The definitions of the competitive ratio for these adversaries can be ac-complished analogously in a straightforward manner, just as it is done for the oblivious adversary. However, the concept of the oblivious adversary is the most widely used concept, and since the focus of this work is the initial establishment of multi-objective online optimization, the adaptive-online adversary and adaptive-offline adversary are not considered in this work.