Epistemic planning is the sub-field of AI planning that focuses on changing knowledge and belief. It is important in both multi-agent domains where agents need to have knowledge/belief regarding the environment, but also the beliefs of other agents, including nested beliefs. When modeling knowledge in multi-agent settings, many models face an exponential growth challenge in terms of nested depth. A contemporary method, known as Planning with Perspectives (PWP), addresses these challenges through the use of perspectives and set operations for knowledge. The JP model defines that an agent's belief is justified if and only if the agent has seen evidence that this belief was true in the past and has not seen evidence to suggest that this has changed. The current paper extends the JP model to handle \emph{group belief}, including distributed belief and common belief. We call this the Group Justified Perspective (GJP) model. Using experimental problems crafted by adapting well-known benchmarks to a group setting, we show the efficiency and expressiveness of our GJP model at handling planning problems that cannot be handled by other epistemic planning tools.

Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.

In this paper, we present a novel approach to epistemic planning called planning with perspectives (PWP) that is both more expressive and computationally more efficient than existing state-of-the-art epistemic planning tools. Epistemic planning — planning with knowledge and belief — is essential in many multi-agent and human-agent interaction domains. Most state-of-the-art epistemic planners solve epistemic planning problems by either compiling to propositional classical planning (for example, generating all possible knowledge atoms or compiling epistemic formulae to normal forms); or explicitly encoding Kripke-based semantics. However, these methods become computationally infeasible as problem sizes grow. In this paper, we decompose epistemic planning by delegating reasoning about epistemic formulae to an external solver. We do this by modelling the problem using Functional STRIPS, which is more expressive than standard STRIPS and supports the use of external, black-box functions within action models. Building on recent work that demonstrates the relationship between what an agent ‘sees’ and what it knows, we define the perspective of each agent using an external function, and build a solver for epistemic logic around this. Modellers can customise the perspective function of agents, allowing new epistemic logics to be defined without changing the planner. We ran evaluations on well-known epistemic planning benchmarks to compare an existing state-of-the-art planner, and on new scenarios that demonstrate the expressiveness of the PWP approach. The results show that our PWP planner scales significantly better than the state-of-the-art planner that we compared against, and can express problems more succinctly.

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable convex relaxations. We invoke the matrix perspective function - the matrix analog of the perspective function-and characterize explicitly the convex hull of epigraphs of convex quadratic, matrix exponential, and matrix power functions under low-rank constraints. Further, we exploit these characterizations to develop strong relaxations for a variety of low-rank problems including reduced rank regression, non-negative matrix factorization, and factor analysis. We establish that these relaxations can be modeled via semidefinite and matrix power cone constraints, and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization, and leads to new relaxations for a broad class of problems.

Epistemic planning --- planning with knowledge and belief --- is essential in many multi-agent and human-agent interaction domains. Most state-of-the-art epistemic planners solve this problem by compiling to propositional classical planning, for example, generating all possible knowledge atoms, or compiling epistemic formula to normal forms. However, these methods become computationally infeasible as problems grow. In this paper, we decompose epistemic planning by delegating reasoning about epistemic formula to an external solver. We do this by modelling the problem using \emph{functional STRIPS}, which is more expressive than standard STRIPS and supports the use of external, black-box functions within action models. Exploiting recent work that demonstrates the relationship between what an agent `sees' and what it knows, we allow modellers to provide new implementations of externals functions. These define what agents see in their environment, allowing new epistemic logics to be defined without changing the planner. As a result, it increases the capability and flexibility of the epistemic model itself, and avoids the exponential pre-compilation step. We ran evaluations on well-known epistemic planning benchmarks to compare with an existing state-of-the-art planner, and on new scenarios based on different external functions. The results show that our planner scales significantly better than the state-of-the-art planner against which we compared, and can express problems more succinctly.