A heuristic framework for a class of robust optimization problems is proposed.The heuristic framework explores dual information.The heuristic is successfully applied to solve two robust optimization problems.The heuristic is able to outperform a widely used 2-approximation procedure.A robust optimization version of the restricted shortest path problem is introduced. This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems.
Later on, Pereira and Averbakh @cite_8 introduced the interval data min-max regret set covering problem. The authors proposed a mathematical formulation for the problem that is similar to the one proposed by @cite_43 for the interval data min-max regret traveling salesman problem. As in @cite_43 , the formulation has an exponential number of constraints. They also adapted the logic-based Benders' algorithm of @cite_32 @cite_43 @cite_40 to this problem and presented an extension of the method that aims at generating multiple Benders' cuts per iteration of the algorithm. Moreover, the work presents an exact approach that uses Benders' cuts in the context of a framework. Computational experiments showed that such approach, as well as the extended logic-based Benders' algorithm, outperforms the standard logic-based Benders' algorithm at solving the instances considered. This robust version of the set covering problem was also addressed in @cite_56 , where the authors propose scenario-based heuristics with path-relinking.
@cite_8 @cite_32 @cite_56 @cite_43 @cite_40
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We apply the exponential weight algorithm, introduced and Littlestone and Warmuth [26]and by Vovk [35]to the problem of predicting a binary sequence almost as well as the best biased coin. We first show that for the case of the logarithmic loss, the derived algorithm is equivalent to the Bayes algorithm with Jeffreys prior, that was studied by Xie and Barron [38]under probabilistic assumptions. We derive a uniform bound on the regret which holds for any sequence. We also show that if the empirical distribution of the sequence is bounded away from 0 and from 1, then, as the length of the sequence increases to infinity, the difference between this bound and a corresponding bound on the average case regret of the same algorithm (which is asymptotically optimal in that case) is only 1 2. We show that this gap of 1 2 is necessary by calculating the regret of the min–max optimal algorithm for this problem and showing that the asymptotic upper bound is tight. We also study the application of this algorithm to the square loss and show that the algorithm that is derived in this case is different from the Bayes algorithm and is better than it for prediction in the worstcase. In the classical stochastic k-armed bandit problem, in each of a sequence of T rounds, a decision maker chooses one of k arms and incurs a cost chosen from an unknown distribution associated with that arm. The goal is to minimize regret, defined as the difference between the cost incurred by the algorithm and the optimal cost. In the linear optimization version of this problem (first considered by Auer [2002]), we view the arms as vectors in R, and require that the costs be linear functions of the chosen vector. As before, it is assumed that the cost functions are sampled independently from an unknown distribution. In this setting, the goal is to find algorithms whose running time and regret behave well as functions of the number of rounds T and the dimensionality n (rather than the number of arms, k, which may be exponential in n or even infinite). We give a nearly complete characterization of this problem in terms of both upper and lower bounds for the regret. In certain special cases (such as when the decision region is a polytope), the regret is polylog(T ). In general though, the optimal regret is Θ∗( √ T ) — our lower bounds rule out the possibility of obtaining polylog(T ) rates in general. We present two variants of an algorithm based on the idea of “upper confidence bounds.” The first, due to Auer [2002], but not fully analyzed, obtains regret whose dependence on n and T are both essentially optimal, but which may be computationally intractable when the decision set is a polytope. The second version can be efficiently implemented when the decision set is a polytope (given as an intersection of half-spaces), but gives up a factor of √ n in the regret bound. Our results also extend to the setting where the set of allowed decisions may change over time. ∗Department of Computer Science, University of Chicago, varsha@cs.uchicago.edu †Toyota Technological Institute at Chicago, hayest,sham @tti-c.org In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward traveling salesman problem (TSP), motivated by robot navigation. In both problems, we are given a graph with lengths on edges and rewards on nodes, and a start node @math . In the Orienteering problem, the goal is to find a path starting at @math that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor @math , and the goal is to maximize the total discounted reward collected, where the reward for a node reached at time @math is discounted by @math . This problem is motivated by an approximation to a planning problem in the Markov decision process (MDP) framework under the commonly employed infinite horizon discounted reward optimality criterion. The approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one-time events and nonrepeatable rewards (such as for package deliveries). We also consider tree and multiple-path variants of these problems and provide approximations for those as well. Although the unrooted Orienteering problem, where there is no fixed start node @math , has been known to be approximable using algorithms for related problems such as @math -TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem in [E. M. Arkin, J. S. B. Mitchell, and G. Narasimhan, Proceedings of the @math th ACM Symposium on Computational Geometry, 1998, pp. 307-316] and [B. Awerbuch, Y. Azar, A. Blum, and S. Vempala, SIAM J. Comput., 28 (1998), pp. 254-262]. We complement our approximation result for Orienteering by showing that the problem is APX-hard. This paper introduces a fast solution procedure to solve 100-node instances of the time-dependent orienteering problem (TD-OP) within a few seconds of computation time. Orienteering problems occur in logistic situations were an optimal combination of locations needs to be selected and the routing between the selected locations needs to be optimized. In the time-dependent variant, the travel time between two locations depends on the departure time at the first location. Next to a mathematical formulation of the TD-OP, the main contribution of this paper is the design of a fast and effective algorithm to tackle this problem. This algorithm combines the principles of an ant colony system (ACS) with a time-dependent local search procedure equipped with a local evaluation metric. Additionally, realistic benchmark instances with varying size and properties are constructed. The average score gap with the known optimal solution on these test instances is only 1.4 with an average computation time of 0.5seconds. An extensive sensitivity analysis shows that the performance of the algorithm is insensitive to small changes in its parameter settings.