In this paper, we present a link between preference-based and multiobjective sequential decision-making. While transforming a multiobjective problem to a preference-based one is quite natural, the other direction is a bit less obvious. We present how this transformation (from preference-based to multiobjective) can be done under the classic condition that preferences over histories can be represented by additively decomposable utilities and that the decision criterion to evaluate policies in a state is based on expectation. This link yields a new source of multiobjective sequential decision-making problems (i.e., when reward values are unknown) and justifies the use of solving methods developed in one setting in the other one.
In multiobjective optimization, four main families of approaches can be distinguished. One first natural approach is to determine the set of all Pareto-optimal solutions (e.g., @cite_9 @cite_26 ). However, in practice, searching for all the Pareto-optimal solutions may not be feasible. Indeed, it is known @cite_22 that this set can be exponential in the size of the state and action spaces. A more practical approach is then to determine an @math -cover of it @cite_0 @cite_22 , which is an approximation of the set of Pareto-optimal solutions. where @math .
@cite_0 @cite_9 @cite_26 @cite_22
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In many multiobjective optimization problems, the Pareto Fronts and Sets contain a large number of solutions and this makes it difficult for the decision maker to identify the preferred ones. A possible way to alleviate this difficulty is to present to the decision maker a subset of a small number of solutions representatives of the Pareto Front characteristics. In this paper, a two-steps procedure is presented, aimed at identifying a limited number of representative solutions to be presented to the decision maker. Pareto Front solutions are first clustered into "families", which are then synthetically represented by a "head-of-the-family" solution. Level Diagrams are then used to represent, analyse and interpret the Pareto Front reduced to its head-of-the-family solutions. The procedure is applied to a reliability allocation case study of literature, in decision-making contexts both without or with explicit preferences by the decision maker on the objectives to be optimized. An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques. We discuss problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives, the so-called Pareto curve. The Pareto curve has typically an exponential number of points. However, it turns out that, under general conditions, there is a polynomially succinct curve that approximates the Pareto curve within any desired accuracy. The central computational question is whether such an approximate curve can be constructed efficiently (in polynomial time). We discuss conditions under which this is the case. We examine in more detail the class of linear multiobjective problems, and relate the multiobjective approximation to the single objective case. We will discuss also problems in multiobjective query optimization. This work deals with bound constrained multiobjective optimization (MOP) of nonsmooth functions for problems where the structure of the objective functions either cannot be exploited, or are absent. Typical situations arise when the functions are computed as the result of a computer simulation. We first present definitions and optimality conditions as well as two families of single-objective formulations of MOP. Next, we propose a new algorithm called for the biobjective optimization (BOP) problem (i.e., MOP with two objective functions). The property that Pareto points may be ordered in BOP and not in MOP is exploited by our algorithm. generates an approximation of the Pareto front by solving a series of single-objective formulations of BOP. These single-objective problems are solved using the recent (mesh adaptive direct search) algorithm for nonsmooth optimization. The Pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the Clarke calculus. Finally, is tested on problems from the literature designed to illustrate specific difficulties encountered in biobjective optimization, such as a nonconvex or disjoint Pareto front, local Pareto fronts, or a nonuniform Pareto front. While Pareto-based multiobjective optimization algorithms continue to show effectiveness for a wide range of practical problems that involve mostly two or three objectives, their limited application for many-objective problems, due to the increasing proportion of nondominated solutions and the lack of sufficient selection pressure, has also been gradually recognized. In this paper, we revive an early developed and computationally expensive strength Pareto-based evolutionary algorithm by introducing an efficient reference direction-based density estimator, a new fitness assignment scheme, and a new environmental selection strategy, for handling both multiobjective and many-objective problems. The performance of the proposed algorithm is validated and compared with some state-of-the-art algorithms on a number of test problems. Experimental studies demonstrate that the proposed method shows very competitive performance on both multiobjective and many-objective problems considered in this paper. Besides, our extensive investigations and discussions reveal an interesting finding, that is, diversity-first-and-convergence-second selection strategies may have great potential to deal with many-objective optimization. For multiobjective optimization problems, it is meaningful to compute a set of solutions covering all possible trade-offs between the different objectives. The multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values. For this problem, efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions, the Pareto frontier, are studied.For the multiobjective one-dimensional knapsack problem, a practical fully polynomial-time approximation scheme (FPTAS) is derived. It is based on a new approach to the single-objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length. For the multiobjectivem-dimensional knapsack problem, the first known polynomial-time approximation scheme (PTAS), based on linear programming, is presented. The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klee's measure problem. In general, Klee's measure problem can be solved with O(n logn + nd 2logn) comparisons for an input instance of size n in d dimensions; as of this writing, it is unknown whether a lower bound higher than Omega(n log n) can be proven. In this paper, we derive a lower bound of Omega(n log n) for the complexity of computing the hypervolume indicator in any number of dimensions d > 1 by reducing the so-called uniformgap problem to it. For the 3-D case, we also present a matching upper bound of O(n log n) comparisons that is obtained by extending an algorithm for finding the maxima of a point set. This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-off among the various conflicting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property which the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational efficiency of continuation-type algorithms. This is an improvement over continuation techniques for tracing the trade-off curve since continuation strategies cannot easily be extended to handle more than two objectives. A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Many classical optimization problems, such as maximum spanning tree and forest, shortest path, maximum weight (perfect) matching, maximum weight independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. Not much is known however about the case of two or more budgets: filling this gap, at least partially, is the main goal of this paper. In more detail, we obtain the following main results: Using iterative rounding for the first time in multi-objective optimization, we obtain multi-criteria PTASs (which slightly violate the budget constraints) for spanning tree, matroid basis, and bipartite matching with @math k = O ( 1 ) budget constraints. We present a simple mechanism to transform multi-criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system. This gives improved algorithms for several problems. In particular, this mechanism can be applied to the above bipartite matching algorithm, hence obtaining a pure PTAS. We show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for @math k -budgeted matroid independent set. We present a deterministic approximation scheme for @math k -budgeted matching (in general graphs), where @math k = O ( 1 ) . Interestingly, to show that our procedure works, we rely on a non-constructive result by Stromquist and Woodall, which is based on the Ham Sandwich Theorem. Evolutionary algorithms (EAs) are often well-suited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EAs are compared quantitatively where an extended 0 1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the strength Pareto EA (SPEA), that combines several features of previous multiobjective EAs in a unique manner. It is characterized by (a) storing nondominated solutions externally in a second, continuously updated population, (b) evaluating an individual's fitness dependent on the number of external nondominated points that dominate it, (c) preserving population diversity using the Pareto dominance relationship, and (d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EAs on the 0 1 knapsack problem. We consider the problem of maximizing a (non-monotone) submodular function subject to a cardinality constraint. In addition to capturing well-known combinatorial optimization problems, e.g., Max-k-Coverage and Max-Bisection, this problem has applications in other more practical settings such as natural language processing, information retrieval, and machine learning. In this work we present improved approximations for two variants of the cardinality constraint for non-monotone functions. When at most k elements can be chosen, we improve the current best 1 e -- o(1) approximation to a factor that is in the range [1 e + 0.004, 1 2], achieving a tight approximation of 1 2 -- o(1) for k = n 2 and breaking the 1 e barrier for all values of k. When exactly k elements must be chosen, our algorithms improve the current best 1 4 -- o(1) approximation to a factor that is in the range [0.356, 1 2], again achieving a tight approximation of 1 2 -- o(1) for k = n 2. Additionally, some of the algorithms we provide are very fast with time complexities of O(nk), as opposed to previous known algorithms which are continuous in nature, and thus, too slow for applications in the practical settings mentioned above. Our algorithms are based on two new techniques. First, we present a simple randomized greedy approach where in each step a random element is chosen from a set of "reasonably good" elements. This approach might be considered a natural substitute for the greedy algorithm of Nemhauser, Wolsey and Fisher [45], as it retains the same tight guarantee of 1--1 e for monotone objectives and the same time complexity of O(nk), while giving an approximation of 1 e for general non-monotone objectives (while the greedy algorithm of Nemhauser et. al. fails to provide any constant guarantee). Second, we extend the double greedy technique, which achieves a tight 1 2 approximation for unconstrained submodular maximization, to the continuous setting. This allows us to manipulate the natural rates by which elements change, thus bounding the total number of elements chosen. In the field of evolutionary computation, there has been a growing interest in applying evolutionary algorithms to solve multimodal optimization problems (MMOPs). Due to the fact that an MMOP involves multiple optimal solutions, many niching methods have been suggested and incorporated into evolutionary algorithms for locating such optimal solutions in a single run. In this paper, we propose a novel transformation technique based on multiobjective optimization for MMOPs, called MOMMOP. MOMMOP transforms an MMOP into a multiobjective optimization problem with two conflicting objectives. After the above transformation, all the optimal solutions of an MMOP become the Pareto optimal solutions of the transformed problem. Thus, multiobjective evolutionary algorithms can be readily applied to find a set of representative Pareto optimal solutions of the transformed problem, and as a result, multiple optimal solutions of the original MMOP could also be simultaneously located in a single run. In principle, MOMMOP is an implicit niching method. In this paper, we also discuss two issues in MOMMOP and introduce two new comparison criteria. MOMMOP has been used to solve 20 multimodal benchmark test functions, after combining with nondominated sorting and differential evolution. Systematic experiments have indicated that MOMMOP outperforms a number of methods for multimodal optimization, including four recent methods at the 2013 IEEE Congress on Evolutionary Computation, four state-of-the-art single-objective optimization based methods, and two well-known multiobjective optimization based approaches. We consider two active binary-classification problems with atypical objectives. In the first, active search, our goal is to actively uncover as many members of a given class as possible. In the second, active surveying, our goal is to actively query points to ultimately predict the proportion of a given class. Numerous real-world problems can be framed in these terms, and in either case typical model-based concerns such as generalization error are only of secondary importance. We approach these problems via Bayesian decision theory; after choosing natural utility functions, we derive the optimal policies. We provide three contributions. In addition to introducing the active surveying problem, we extend previous work on active search in two ways. First, we prove a novel theoretical result, that less-myopic approximations to the optimal policy can outperform more-myopic approximations by any arbitrary degree. We then derive bounds that for certain models allow us to reduce (in practice dramatically) the exponential search space required by a naive implementation of the optimal policy, enabling further lookahead while still ensuring that optimal decisions are always made. We study problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives (the so-called Pareto curve). We point out that, under very general conditions, there is a polynomially succinct curve that spl epsiv -approximates the Pareto curve, for any spl epsiv >0. We give a necessary and sufficient condition under which this approximate Pareto curve can be constructed in time polynomial in the size of the instance and 1 spl epsiv . In the case of multiple linear objectives, we distinguish between two cases: when the underlying feasible region is convex, then we show that approximating the multi-objective problem is equivalent to approximating the single-objective problem. If however the feasible region is discrete, then we point out that the question reduces to an old and recurrent one: how does the complexity of a combinatorial optimization problem change when its feasible region is intersected with a hyperplane with small coefficients; we report some interesting new findings in this domain. Finally, we apply these concepts and techniques to formulate and solve approximately a cost-time-quality trade-off for optimizing access to the World-Wide Web, in a model first studied by (1996) (which was actually the original motivation for this work). This paper is devoted to fair optimization in Multiobjective Markov Decision Processes (MOMDPs). A MOMDP is an extension of the MDP model for planning under uncertainty while trying to optimize several reward functions simultaneously. This applies to multiagent problems when rewards define individual utility functions, or in multicriteria problems when rewards refer to different features. In this setting, we study the determination of policies leading to Lorenz-non-dominated tradeoffs. Lorenz dominance is a refinement of Pareto dominance that was introduced in Social Choice for the measurement of inequalities. In this paper, we introduce methods to efficiently approximate the sets of Lorenz-non-dominated solutions of infinite-horizon, discounted MOMDPs. The approximations are polynomial-sized subsets of those solutions. We propose a noniterative solution for the Perspective-n-Point ( P n P ) problem, which can robustly retrieve the optimum by solving a seventh order polynomial. The central idea consists of three steps: 1) to divide the reference points into 3-point subsets in order to achieve a series of fourth order polynomials, 2) to compute the sum of the square of the polynomials so as to form a cost function, and 3) to find the roots of the derivative of the cost function in order to determine the optimum. The advantages of the proposed method are as follows: First, it can stably deal with the planar case, ordinary 3D case, and quasi-singular case, and it is as accurate as the state-of-the-art iterative algorithms with much less computational time. Second, it is the first noniterative P n P solution that can achieve more accurate results than the iterative algorithms when no redundant reference points can be used (n 5). Third, large-size point sets can be handled efficiently because its computational complexity is O(n). In this paper we present the PUMP (Parallel Uncertainty-aware Multiobjective Planning) algorithm for addressing the stochastic kinodynamic motion planning problem, whereby one seeks a low-cost, dynamically-feasible motion plan subject to a constraint on collision probability (CP). To ensure exhaustive evaluation of candidate motion plans (as needed to tradeoff the competing objectives of performance and safety), PUMP incrementally builds the Pareto front of the problem, accounting for the optimization objective and an approximation of CP. This is performed by a massively parallel multiobjective search, here implemented with a focus on GPUs. Upon termination of the exploration phase, PUMP searches the Pareto set of motion plans to identify the lowest cost solution that is certified to satisfy the CP constraint (according to an asymptotically exact estimator). We introduce a novel particle-based CP approximation scheme, designed for efficient GPU implementation, which accounts for dependencies over the history of a trajectory execution. We present numerical experiments for quadrotor planning wherein PUMP identifies solutions in 100 ms, evaluating over one hundred thousand partial plans through the course of its exploration phase. The results show that this multiobjective search achieves a lower motion plan cost, for the same CP constraint, compared to a safety buffer-based search heuristic and repeated RRT trials. We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. We introduce (1 + λ) and (μ | λ) variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks. Motivated by multi-budgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution x in the (non-bipartite graph) matching and matroid intersection polytopes. We show that for any fixed Δ > 0, a given point x can be rounded to a random solution R such that E[1R] = (1 − Δ)x and any linear function of x satisfies dimension-free Chernoff-Hoeffding concentration bounds (the bounds depend on Δ and the expectation μ). We build on and adapt the swap rounding scheme in our recent work [9] to achieve this result. Our main contribution is a non-trivial martingale based analysis framework to prove the desired concentration bounds. In this paper we describe two applications. We give a randomized PTAS for matroid intersection and matchings with any fixed number of budget constraints. We also give a deterministic PTAS for the case of matchings. The concentration bounds also yield related results when the number of budget constraints is not fixed. As a second application we obtain an algorithm to compute in polynomial time an ∈-approximate Pareto-optimal set for the multi-objective variants of these problems, when the number of objectives is a fixed constant. We rely on a result of Papadimitriou and Yannakakis [26]. We study the classic mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single parameter settings (i.e., where each agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [20]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [25]. In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. We prove that these mechanisms are approximately optimal in single-dimensional settings. These posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical. Furthermore, these mechanisms generalize naturally to multi-dimensional settings where they give the first known approximations to the elusive optimal multi-dimensional mechanism design problem. In particular, we solve multi-dimensional multi-unit auction problems and generalizations to matroid feasibility constraints. The constant approximations we obtain range from 1.5 to 8. For all but one case, our posted price sequences can be computed in polynomial time. This work can be viewed as an extension and improvement of the single-agent algorithmic pricing work of [9] to the setting of multiple agents where the designer has combinatorial feasibility constraints on which agents can simultaneously obtain each service.