Choosing a diverse subset of a large collection of points in a metric space is a fundamental problem, with applications in feature selection, recommender systems, web search, data summarization, etc. Various notions of diversity have been proposed, tailored to different applications. The general algorithmic goal is to find a subset of points that maximize diversity, while obeying a cardinality (or more generally, matroid) constraint. The goal of this paper is to develop a novel linear programming (LP) framework that allows us to design approximation algorithms for such problems. We study an objective known as {\em sum-min} diversity, which is known to be effective in many applications, and give the first constant factor approximation algorithm. Our LP framework allows us to easily incorporate additional constraints, as well as secondary objectives. We also prove a hardness result for two natural diversity objectives, under the so-called {\em planted clique} assumption. Finally, we study the empirical performance of our algorithm on several standard datasets.

Such mistakes can have disastrous and even potentially fatal consequences.

In Appendix B, we provide proofs of the theorems. In Table 6, we provide a list of oracle functions of three basic operation types, including affine transformation, unary nonlinear function, and binary nonlinear function. This lower bound can be used for training ReLU networks with loss fusion. Figure 4 compares the linear bounds in LiRP A and IBP respesctively. We refer readers to those existing works for details.

Recall that as introduced in Section 3.3, given bounds of The lower bound can be any line with a slope between 0 and 1 and a bias of 0, i.e., We illustrate the linear relaxation in Figure 4. A.2 Linear Relaxation for Absolute V alue Function In Figure 5, we illustrate the linear relaxation for the absolute value function. Settings other than the models are the same as those for experiments in Table 1. In this section, we show additional empirical results on the synthetic dataset. For deeper models, our method outperforms previous works with larger gaps. We show the results in Table 7.

Although the provided novel algorithm looks impressive both from the theoretical prospective and in the experimental comparison, its substantiation has quite some room for improvement. The major point is the proof of Theorem 1: - it is unclear how the proof of the theorem follows from Lemmas 3 and 4, since none of these lemmas is related to the optimal solution of the considered diversity problem. I assume that the missing proposition is the one, which would establish connection between the considered linear program in lines 153-154 (by the way, it is very uncomfortable that the main formulation is not numbered and therefore can not be easily referenced) and the diversity problem. I believe that this connection may have the following format: if the linear program is equipped with integrality constraints (which is, all variables x_{ir}\in {0,1}), the resulting ILP is equivalent to the considered diversity problem. Indeed, the proof of such a proposition is not obvious for me as well.

Branch-and-bound (BaB) is among the most effective techniques for neural network (NN) verification. However, existing works on BaB for NN verification have mostly focused on NNs with piecewise linear activations, especially ReLU networks. In this paper, we develop a general framework, named GenBaB, to conduct BaB on general nonlinearities to verify NNs with general architectures, based on linear bound propagation for NN verification. To decide which neuron to branch, we design a new branching heuristic which leverages linear bounds as shortcuts to efficiently estimate the potential improvement after branching. To decide nontrivial branching points for general nonlinear functions, we propose to pre-optimize branching points, which can be efficiently leveraged during verification with a lookup table. We demonstrate the effectiveness of our GenBaB on verifying a wide range of NNs, including NNs with activation functions such as Sigmoid, Tanh, Sine and GeLU, as well as NNs involving multi-dimensional nonlinear operations such as multiplications in LSTMs and Vision Transformers. Our framework also allows the verification of general nonlinear computation graphs and enables verification applications beyond simple NNs, particularly for AC Optimal Power Flow (ACOPF). GenBaB is part of the latest $\alpha,\!\beta$-CROWN, the winner of the 4th and the 5th International Verification of Neural Networks Competition (VNN-COMP 2023 and 2024).

Although nearly 20 years have passed since its conception, the feasibility pump algorithm remains a widely used heuristic to find feasible primal solutions to mixed-integer linear problems. Many extensions of the initial algorithm have been proposed. Yet, its core algorithm remains centered around two key steps: solving the linear relaxation of the original problem to obtain a solution that respects the constraints, and rounding it to obtain an integer solution. This paper shows that the traditional feasibility pump and many of its follow-ups can be seen as gradient-descent algorithms with specific parameters. A central aspect of this reinterpretation is observing that the traditional algorithm differentiates the solution of the linear relaxation with respect to its cost. This reinterpretation opens many opportunities for improving the performance of the original algorithm. We study how to modify the gradient-update step as well as extending its loss function. We perform extensive experiments on MIPLIB instances and show that these modifications can substantially reduce the number of iterations needed to find a solution.

This paper presents a model predictive control (MPC) for dynamic systems whose nonlinearity and uncertainty are modelled by deep neural networks (NNs), under input and state constraints. Since the NN output contains a high-order complex nonlinearity of the system state and control input, the MPC problem is nonlinear and challenging to solve for real-time control. This paper proposes two types of methods for solving the MPC problem: the mixed integer programming (MIP) method which produces an exact solution to the nonlinear MPC, and linear relaxation (LR) methods which generally give suboptimal solutions but are much computationally cheaper. Extensive numerical simulation for an inverted pendulum system modelled by ReLU NNs of various sizes is used to demonstrate and compare performance of the MIP and LR methods.