The aim of this paper is to propose a new approach interweaving preference elicitation and search to solve multiobjective optimization problems. We present an interactive search procedure directed by an aggregation function, possibly non-linear (e.g. an additive disutility function, a Choquet integral), defining the overall cost of solutions. This function is parameterized by weights that are initially unknown. Hence, we insert comparison queries in the search process to obtain useful preference information that will progressively reduce the uncertainty attached to weights. The process terminates by recommending a near-optimal solution ensuring that the gap to optimality is below the desired threshold. Our approach is tested on multiobjective state space search problems and appears to be quite efficient both in terms of number of queries and solution times.

Continuous optimization is an important problem in many areas of AI, including vision, robotics, probabilistic inference, and machine learning. Unfortunately, most real-world optimization problems are nonconvex, causing standard convex techniques to find only local optima, even with extensions like random restarts and simulated annealing. We observe that, in many cases, the local modes of the objective function have combinatorial structure, and thus ideas from combinatorial optimization can be brought to bear. Based on this, we propose a problem-decomposition approach to nonconvex optimization. Similarly to DPLL-style SAT solvers and recursive conditioning in probabilistic inference, our algorithm, RDIS, recursively sets variables so as to simplify and decompose the objective function into approximately independent sub-functions, until the remaining functions are simple enough to be optimized by standard techniques like gradient descent. The variables to set are chosen by graph partitioning, ensuring decomposition whenever possible. We show analytically that RDIS can solve a broad class of nonconvex optimization problems exponentially faster than gradient descent with random restarts. Experimentally, RDIS outperforms standard techniques on problems like structure from motion and protein folding.

We consider the problem of generating optimal stochastic policies for Constrained Stochastic Shortest Path problems, which are a natural model for planning under uncertainty for resource-bounded agents with multiple competing objectives. While unconstrained SSPs enjoy a multitude of efficient heuristic search solution methods with the ability to focus on promising areas reachable from the initial state, the state of the art for constrained SSPs revolves around linear and dynamic programming algorithms which explore the entire state space. In this paper, we present i-dual, which, to the best of our knowledge, is the first heuristic search algorithm for constrained SSPs. To concisely represent constraints and efficiently decide their violation, i-dual operates in the space of dual variables describing the policy occupation measures. It does so while retaining the ability to use standard value function heuristics computed by well-known methods. Our experiments on a suite of PPDDL problems augmented with constraints show that these features enable i-dual to achieve up to two orders of magnitude improvement in run-time and memory over linear programming algorithms.

In this work, we investigate a multi-period Home Health Care Scheduling Problem (HHCSP) under stochastic service and travel times. We first model the deterministic problem as an integer linear programming model that incorporates real-world requirements, such as time windows, continuity of care, workload fairness, inter-visit temporal dependencies. We then extend the model to cope with uncertainty in durations, by introducing chance constraints into the formulation. We propose efficient solution approaches, which provide quantifiable near-optimal solutions and further handle the uncertainties by employing a sampling-based strategy. We demonstrate the effectiveness of our proposed approaches on instances synthetically generated by real-world dataset for both deterministic and stochastic scenarios.

For example, we may need to (i) perform inverse learning of the cost parameters of a multi-objective reward based on observed agent behavior; (ii) perform sensitivity analyses of policies to various parameter settings; or (iii) analyze and optimize policy performance as a function of policy parameters. When such problems have mixed discrete and continuous state and/or action spaces, this leads to parameterized hybrid MDPs (PHMDPs) that are often approximately solved via discretization, sampling, and/or local gradient methods (when optimization is involved). In this paper we combine two recent advances that allow for the first exact solution and optimization of PHMDPs. We first show how each of the aforementioned use cases can be formalized as PHMDPs, which can then be solved via an extension of symbolic dynamic programming (SDP) even when the solution is piecewise nonlinear. Secondly, we can leverage recent advances in non-convex solvers that require symbolic forms of the objective function for non-convex global optimization in (i), (ii), and (iii) using SDP to derive symbolic solutions for each PHMDP formalization. We demonstrate the efficacy and scalability of our optimal analytical framework on nonlinear examples of each of the aforementioned use cases.

Last-mile transportation (LMT) refers to any service that moves passengers from a hub of mass transportation (MT), such as air, boat, bus, or train, to destinations, such as a home or an office. In this paper, we introduce the problem of scheduling passengers jointly on MT and LMT services, with passengers sharing a car, van, or autonomous pod of limited capacity for LMT. Passenger itineraries are determined so as to minimize total transit time for all passengers, with each passenger arriving at the destination within a specified time window. The transit time includes the time spent traveling through both services and, possibly, waiting time for transferring between the services. We provide an integer linear programming (ILP) formulation for this problem. Since the ILMTP, is NP-hard and problem instances of practical size are often difficult to solve, we study a restricted version where MT trips are uniform, all passengers have time windows of a common size, and LMT vehicles visit one destination per trip. We prove that there is an optimal solution that sorts and groups passengers by their deadlines and, based on this result, we propose a constructive grouping heuristic and local search operators to generate high-quality solutions. The resulting groups are optimally scheduled in a few seconds using another ILP formulation. Numerical results indicate that the solutions obtained by this heuristic are often close to optimal %, even when multiple destinations are allowed per group, and that warm-starting the ILP solver with such solutions decreases the overall computational times significantly.

Abduction is a form of inference that seeks the best explanation for the given observation. Because it provides a reasoning process based on background knowledge, it is used in applications that need convincing explanations. In this study, we consider weighted abduction, which is one of the commonly used mathematical models for abduction. The main difficulty associated with applying weighted abduction to real problems is its computational complexity. A state-of-the-art method formulates weighted abduction as an integer linear programming (ILP) problem and solves it using efficient ILP solvers; however, it is still limited to solving problems that include at most 100 rules of background knowledge and observations.

Heuristic search planning effectively finds solutions for large planning problems, but since the estimates are either not admissible or too weak, optimal solutions are found in rare cases only. In contrast, heuristic pattern databases are known to significantly improve lower bound estimates for optimally solving challenging single-agent problems like the 24-Puzzle or Rubik's Cube. This paper studies the effect of pattern databases in the context of deterministic planning. Given a fixed state description based on instantiated predicates, we provide a general abstraction scheme to automatically create admissible domain-independent memory-based heuristics for planning problems, where abstractions are found in factorizing the planning space. We evaluate the impact of pattern database heuristics in A* and hill climbing algorithms for a collection of benchmark domains.

Low-rank and nonsmooth matrix optimization problems capture many fundamental tasks in statistics and machine learning. While significant progress has been made in recent years in developing efficient methods for \textit{smooth} low-rank optimization problems that avoid maintaining high-rank matrices and computing expensive high-rank SVDs, advances for nonsmooth problems have been slow paced. In this paper we consider standard convex relaxations for such problems. Mainly, we prove that under a natural \textit{generalized strict complementarity} condition and under the relatively mild assumption that the nonsmooth objective can be written as a maximum of smooth functions, the \textit{extragradient method}, when initialized with a "warm-start" point, converges to an optimal solution with rate $O(1/t)$ while requiring only two \textit{low-rank} SVDs per iteration. We give a precise trade-off between the rank of the SVDs required and the radius of the ball in which we need to initialize the method. We support our theoretical results with empirical experiments on several nonsmooth low-rank matrix recovery tasks, demonstrating that using simple initializations, the extragradient method produces exactly the same iterates when full-rank SVDs are replaced with SVDs of rank that matches the rank of the (low-rank) ground-truth matrix to be recovered.

Kernel-based models such as kernel ridge regression and Gaussian processes are ubiquitous in machine learning applications for regression and optimization. It is well known that a serious downside for kernel-based models is the high computational cost; given a dataset of $n$ samples, the cost grows as $\mathcal{O}(n^3)$. Existing sparse approximation methods can yield a significant reduction in the computational cost, effectively reducing the real world cost down to as low as $\mathcal{O}(n)$ in certain cases. Despite this remarkable empirical success, significant gaps remain in the existing results for the analytical confidence bounds on the error due to approximation. In this work, we provide novel confidence intervals for the Nystr\"om method and the sparse variational Gaussian processes approximation method. Our confidence intervals lead to improved error bounds in both regression and optimization. We establish these confidence intervals using novel interpretations of the approximate (surrogate) posterior variance of the models.