An Algebraic Perspective of Group Relaxations

一阶SDSOS-凸半代数优化与精确SOCP松弛

First-order SDSOS-convex semi-algebraic optimization and exact SOCP relaxations

本文提出一阶SDSOS-凸半代数函数，推广了多项式情形，并证明其规划问题可通过精确求解二阶锥规划获得最优解，应用于鲁棒优化。

Gaining or Losing Perspective for Convex Multivariate Functions on a Simplex

MINLO (mixed-integer nonlinear optimization) formulations of the disjunction between the origin and a polytope via a binary indicator variable have broad applicability in nonlinear combinatorial optimization, for modeling a fixed cost $c$ associated with carrying out a set of $d$ activities and a convex variable cost function $f$ associated with the levels of the activities. The perspective relaxation is often used to solve such models to optimality in a branch-and-bound context, especially in the context in which $f$ is univariate (e.g., in Markowitz-style portfolio optimization). But such a …

数据驱动预测控制中的正则化：一种凸松弛视角

Regularization in Data-driven Predictive Control: A Convex Relaxation Perspective

本文通过凸松弛视角探讨数据驱动预测控制中正则化的作用，提出A-DDPC方法，通过迭代算法解决内层问题，实验表明其在减少偏差和方差误差方面优于现有方法。

Optimal control of differential-algebraic equations from an ordinary differential equation perspective

We study the Optimal Control Problem (OCP) for regular linear differential-algebraic systems (DAEs). To this end, we introduce the input index, which allows, on the one hand, to characterize the space of consistent initial values in terms of a Kalman-like matrix and, on the other hand, the necessary smoothness properties of the control. The latter is essential to make the problem accessible from a numerical point of view. Moreover, we derive an augmented system as the key to analyze the OCP with tools well-known from optimal control of ordinary differential equations. The new concepts of the i…

线性代数、优化与统计中的格拉斯曼簇与旗簇：代数视角

Grassmann and Flag Varieties in Linear Algebra, Optimization, and Statistics: An Algebraic Perspective

本文从代数视角研究格拉斯曼簇与旗簇，将线性代数与统计问题转化为其上的优化问题，以复临界点数量衡量代数复杂度，阐述流形的代数簇实现并计算了多类优化问题的代数复杂度。

正则化与松弛：从锥优化视角看统计变量选择

Regularization vs. Relaxation: A conic optimization perspective of statistical variable selection

本文从锥优化视角探讨变量选择问题，证明MCP和反Huber惩罚函数可视为视角松弛的特例，并通过半定松弛解决，结合Goemans-Williamson方法获得近似解。

Exploiting Sparsity for Semi-Algebraic Set Volume Computation

We provide a systematic deterministic numerical scheme to approximate the volume (i.e. the Lebesgue measure) of a basic semi-algebraic set whose description follows a sparsity pattern. As in previous works (without sparsity), the underlying strategy is to consider an infinite-dimensional linear program on measures whose optimal value is the volume of the set. This is a particular instance of a generalized moment problem which in turn can be approximated as closely as desired by solving a hierarchy of semidefinite relaxations of increasing size. The novelty with respect to previous work is that…

Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness

The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that, for convex polynomial optimization, the Lasserre hierarchy with a slightly extended quadratic module always converges asymptotically even in the face of non-compact semi-algebraic feasible sets. We do this by exploiting a coercivity property of convex polynomials that are bounded below. We further establish that the positive definiteness of the Hessian of the …

低秩优化的新视角

A new perspective on low-rank optimization

本研究引入矩阵透视函数，结合正交投影矩阵提出矩阵透视重构技术，可高效生成各类低秩问题的强凸松弛，推广了混合整数优化的透视重构方法。

在箱型域上为凸多元函数获得或失去透视

Gaining or losing perspective for convex multivariate functions on box domains

本文研究箱型域上凸多元函数的透视松弛，以体积衡量紧致性，并借助三角剖分构造最小凹上界，将单纯形情形的结果推广到箱型情形。

A feasible method for solving an SDP relaxation of the quadratic knapsack problem

In this paper, we consider an SDP relaxation of the quadratic knapsack problem (QKP). After using the Burer-Monteiro factorization, we get a non-convex optimization problem, whose feasible region is an algebraic variety. Although there might be non-regular points on the algebraic variety, we prove that the algebraic variety is a smooth manifold except for a trivial point for a generic input data. We also analyze the local geometric properties of non-regular points on this algebraic variety. In order to maintain the equivalence between the SDP problem and its non-convex formulation, we derive a…

Improved Structural Methods for Nonlinear Differential-Algebraic Equations via Combinatorial Relaxation

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Exi…

SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with $\ell_1$-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex…

A Computational Study of Perspective Cuts

The benefits of cutting planes based on the perspective function are well known for many specific classes of mixed-integer nonlinear programs with on/off structures. However, we are not aware of any empirical studies that evaluate their applicability and computational impact over large, heterogeneous test sets in general-purpose solvers. This paper provides a detailed computational study of perspective cuts within a linear programming based branch-and-cut solver for general mixed-integer nonlinear programs. Within this study, we extend the applicability of perspective cuts from convex to nonco…

The equivalence of optimal perspective formulation and Shor's SDP for quadratic programs with indicator variables

In this paper, we compare the strength of the optimal perspective reformulation and Shor's SDP relaxation. We prove these two formulations are equivalent for quadratic optimization problems with indicator variables.

Gaining or losing perspective

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[l,u]$, where $z$ is a binary indicatorof $x\in[l,u]$ ($u> \ell > 0$), and $y$ "captures" $f(x)$, which is assumed to be convex on its domain $[l,u]$, but otherwise $y=0$ when $x=0$. This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achi…

Comparing perspective reformulations for piecewise-convex optimization

Our study is motivated by the solution of Mixed-Integer Non-Linear Programming (MINLP) problems with separable non-convex functions via the Sequential Convex MINLP technique, an iterative method whose main characteristic is that of solving, for bounding purposes, piecewise-convex MINLP relaxations obtained by identifying the intervals in which each univariate function is convex or concave and then relaxing the concave parts with piecewise-linear relaxations of increasing precision. This process requires the introduction of new binary variables for the activation of the intervals where the func…

Gaining or Losing Perspective for Piecewise-Linear Under-Estimators of Convex Univariate Functions

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[\ell,u]$, where $z$ is a binary indicator of $x\in[\ell,u]$ ($0 \leq \ell <u$), and $y$ "captures" $f(x)$, which is assumed to be convex and positive on its domain $[\ell,u]$, but otherwise $y=0$ when $x=0$. This model is very useful in nonlinear combinatorial optimization, where there is a fixed cost of operating an activity at level $x$ in the operating range $[\ell,u]$, and then there is a further (convex) variable cost $f(x)$. In particular, we study relaxations related to the perspective t…

基于变量聚合的视角重构在具有对称性的混合整数凸优化中的应用

Variable Aggregation-based Perspective Reformulation for Mixed-Integer Convex Optimization with Symmetry

本文提出一种基于变量聚合和视角重构的方法，用于解决具有对称性的混合整数凸优化问题，通过精确刻画可行区域的凸包来提升松弛效果。

投影到带帽旋转二阶锥

Projecting onto a Capped Rotated Second-Order Cone

推导了带帽旋转二阶锥投影的闭式表达式，涉及三种情况，并给出非平凡投影解位于锥与盒子面交点的充要条件。

由多项式矩阵不等式定义的半代数集的非 SOS 正零点定理

Non-SOS Positivstellensätze for semi-algebraic sets defined by polynomial matrix inequalities

本文将 Handelman 与 Krivine-Stengle 定理推广至多项式矩阵不等式，提出非 SOS 的正性证书及稀疏表示，并构建规模不随变量数增长的半定规划松弛层级。

一种用于混合二元二次规划双重非负松弛的低秩增广拉格朗日方法

A low-rank augmented Lagrangian method for doubly nonnegative relaxations of mixed-binary quadratic programs

本文提出全局收敛的黎曼低秩增广拉格朗日方法 RNNAL，通过等价简化 BM 分解约束并求解非凸代数簇投影，高效处理大规模混合二元二次规划的 DNN 松弛。

D-optimal design for multivariate polynomial regression via the Christoffel function and semidefinite relaxations

We present a new approach to the design of D-optimal experiments with multivariate polynomial regressions on compact semi-algebraic design spaces. We apply the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically and approximately the optimal design problem. The geometry of the design is recovered with semidefinite programming duality theory and the Christoffel polynomial.

Semidefinite Programming for Min-Max Problems and Games

We introduce two min-max problems: the first problem is to minimize the supremum of finitely many rational functions over a compact basic semi-algebraic set whereas the second problem is a 2-player zero-sum polynomial game in randomized strategies and with compact basic semi-algebraic pure strategy sets. It is proved that their optimal solution can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre. This provides a unified approach and a class of algorithms to approximate all Nash equilibria and min-max strategies of m…

Algebraic perspectives on signomial optimization

Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of ``degree'' that is central to polynomial optimization theory. We reclaim that principle here through the concept of signomial rings, which we use to derive complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. The Positivstellensatz underlying the lower bounds relies on the conc…

Multi-objective convex polynomial optimization and semidefinite programming relaxations

This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem (P$_z$) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem (P$_z$), we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite conve…

Qualification Conditions in Semi-algebraic Programming

For an arbitrary finite family of semi-algebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies Mangasarian-Fromovitz constraint qualification. Using the Milnor-Thom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential…

Euclidean Distance degrees of real algebraic groups

We study the problem of finding, in a real algebraic matrix group, the matrix closest to a given data matrix. We do so from the algebro-geometric perspective of Euclidean distance degrees. We recover several classical results; and among the new results that we prove is a formula for the Euclidean distance degree of special linear groups.

Minimizing Polynomial Functions

We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gröbner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming. This opens up the possibility of using semidefinite programming relaxations arising from the Positivstellensatz for a wide range of computational problems in real algebraic geometry. This paper was presented at the Workshop on Algorithmic and Quantitative A…

Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, e.g., algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [9] present an SDP based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program, but generally dominates it on small instances. We provide a family of \emp…