Minimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of Squares

大数据中最小体积覆盖椭球问题的高效数据驱动杠杆得分采样算法

Efficient Data-Driven Leverage Score Sampling Algorithm for the Minimum Volume Covering Ellipsoid Problem in Big Data

针对大数据中最小体积覆盖椭球问题，提出数据驱动杠杆得分采样算法，将计算复杂度从O(nd^2)降至O(nd+poly(d))，并保证收敛与近最优解。

A generalization of Löwner-John's ellipsoid theorem

We address the following generalization $P$ of the Lowner-John ellipsoid problem. Given a (non necessarily convex) compact set $K\subset R^n$ and an even integer $d$, find an homogeneous polynomial $g$ of degree $d$ such that $K\subset G:=\{x:g(x)\leq1\}$ and $G$ has minimum volume among all such sets. We show that $P$ is a convex optimization problem even if neither $K$ nor $G$ are convex! We next show that $P$ has a unique optimal solution and a characterization with at most ${n+d-1\choose d}$ contacts points in $K\cap G$ is also provided. This is the analogue for $d\textgreater{}2$ of the L…

可定义椭球法、平方和证明与图同构问题

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Graph Isomorphism Problem

本文指出带计数的定点逻辑（FPC）中的椭球法归约适用于显式有界凸集族，证明半正定规划精确可行性可用无穷FPC表达，并得出图同构问题中Lasserre/平方和层级会塌缩至Sherali-Adams层级的结论。

包含p-和的最小椭球体的椭球体边界问题及其在可达性分析中的应用

Smallest Ellipsoid Containing $p$-Sum of Ellipsoids with Application to Reachability Analysis

本文研究了凸集值数据的椭球体边界问题，其中凸集由有限多个椭球体的p-和构成，对于任何实数p≥1。p-和的概念出现在凸分析的Brunn-Minkowski-Firey理论中，并推广了已知的集值运算，如Minkowski和。我们推导了给定椭球体集合的p-和的外椭球参数化，并计算了两种最优性标准下的最紧参数化：最小迹和最小体积。对于这些最优参数化，系统控制文献中的几个已知结果作为我们一般公式的特殊情况被恢复。对于最小体积标准，我们的分析导致了一个标量参数化外椭球形状矩阵的固定点递归。该递归被证明是合同性的，并且在实践中收敛迅速。我们将这些结果应用于计算线性控制系统在不同凸集值不确定性模型下的前向可达集，这些模型由变化的p∈[1,∞]生成。我们的数值结果表明，所提出的固定点算法在p=1时相比现有的半正定规划方法，计算时间快了超过两个数量级，而对数值精度影响不大。对于p>1，这里报告的可达集计算结果是新的。我们的结果预计在实时安全关键应用中很有用，如自动驾驶车辆的碰撞避免决策，其中可达集计算的时间尺度需要远小于车辆动力学时间尺度。

Linear minimum mean square filters for Markov jump linear systems

New linear minimum mean square estimators are introduced in this paper by considering a cluster information structure in the filter design. The set of filters constructed in this way can be ordered in a lattice according to the refines of clusters of the Markov chain, including the linear Markovian estimator at one end (with only one cluster) and the Kalman filter at the other hand (with as many clusters as Markov states). The higher is the number of clusters, the heavier are pre-compuations and smaller is the estimation error, so that the cluster cardinality allows for a trade-off between per…

On Some Incremental Algorithms for the Minimum Sum-of-Squares Clustering Problem. Part 1: Ordin and Bagirov's Incremental Algorithm

Solution methods for the minimum sum-of-squares clustering (MSSC) problem are analyzed and developed in this paper. Based on the DCA (Difference-of-Convex functions Algorithms) in DC programming and recently established qualitative properties of the MSSC problem, we suggest several improvements of the incremental algorithms of Ordin and Bagirov and of Bagirov. Properties of the new algorithms are obtained and preliminary numerical tests of those on real-world databases are shown. Finite convergence, convergence, and the rate of convergence of solution methods for the MSSC problem are presented…

可扩展的矩阵椭球体Minkowski和的外近似用于数据驱动控制

Scalable Outer Approximation of Minkowski Sums of Matrix Ellipsoids for Data-Driven Control

本文提出一种无需线性矩阵不等式的方法，用于高效计算数据驱动控制中多个矩阵椭球体Minkowski和的外近似，通过分析半轴平方和体积两种标准，提升计算效率和可扩展性。

SOS-SDP: an Exact Solver for Minimum Sum-of-Squares Clustering

The minimum sum-of-squares clustering problem (MSSC) consists of partitioning $n$ observations into $k$ clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper, we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure where valid inequalities are iteratively added to the Peng-Wei SDP relaxation. The upper bound is computed with the constrained version of k-means where the initial centroids are extracted from the solution of the SDP…

Qualitative Properties of the Minimum Sum-of-Squares Clustering Problem

A series of basic qualitative properties of the minimum sum-of-squares clustering problem are established in this paper. Among other things, we clarify the solution existence, properties of the global solutions, characteristic properties of the local solutions, locally Lipschitz property of the optimal value function, locally upper Lipschitz property of the global solution map, and the Aubin property of the local solution map. We prove that the problem in question always has a global solution and, under a mild condition, the global solution set is finite and the components of each global solut…

Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is …

Efficient Algorithms for Positive Semi-Definite Total Least Squares Problems, Minimum Rank Problem and Correlation Matrix Computation

We have recently presented a method to solve an overdetermined linear system of equations with multiple right hand side vectors, where the unknown matrix is to be symmetric and positive definite. The coefficient and the right hand side matrices are respectively named data and target matrices. A more complicated problem is encountered when the unknown matrix is to be positive semi-definite. The problem arises in estimating the compliance matrix to model deformable structures and approximating correlation and covariance matrices in financial modeling. Several methods have been proposed for solvi…

关于计算统计学关键问题的稀疏通用求解器

On computing sparse universal solvers for key problems in statistics

本文针对最小二乘相关问题的通用求解器，提出稀疏性结果与算法，包括最小1-范数、最小秩及同时最小范数/最小二乘的通用求解器，并比较了多种线性规划公式与近端点算法，其中Douglas-Rachford分裂算法表现最佳。

Kalman-Bucy filtering and minimum mean square estimator under uncertainty

In this paper, we study a generalized Kalman-Bucy filtering problem under uncertainty. The drift uncertainty for both signal process and observation process is considered and the attitude to uncertainty is characterized by a convex operator (convex risk measure). The optimal filter or the minimum mean square estimator (MMSE) is calculated by solving the minimum mean square estimation problem under a convex operator. In the first part of this paper, this estimation problem is studied under g-expectation which is a special convex operator. For this case, we prove that there exists a worst-case p…

最小方形铺砌的稀疏角差MILP

A Sparse Corner-Difference MILP for Minimum Square Tiling

针对最小方形铺砌问题，提出角差MILP（CD-MILP），将非零系数从Θ(N^5)降至Θ(N^3)，并引入角排序约束处理对称性，实验表明组合方法显著提升求解性能。

一种用于最小平方和聚类的动态约束聚合列生成算法

A column generation algorithm with dynamic constraint aggregation for minimum sum-of-squares clustering

本文针对大规模最小平方和聚类问题，提出将列生成与动态约束聚合相结合的精确求解算法，通过消融研究优化设计，显著超越现有最先进精确方法。

The Importance of Good Starting Solutions in the Minimum Sum of Squares Clustering Problem

The clustering problem has many applications in Machine Learning, Operations Research, and Statistics. We propose three algorithms to create starting solutions for improvement algorithms for this problem. We test the algorithms on 72 instances that were investigated in the literature. Forty eight of them are relatively easy to solve and we found the best known solution many times for all of them. Twenty four medium and large size instances are more challenging. We found five new best known solutions and matched the best known solution for 18 of the remaining 19 instances.

An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering

The minimum sum-of-squares clustering (MSSC), or k-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote interpretability of the clustering process has become a hot research topic at the intersection of mathematical optimization and machine learning research. The problem of taking advantage of background information in data clustering is called semi-supervised or constrained clustering. In this paper, we present a branch-and-cut algorithm for semi-supervised MSSC, where backg…