Linear programs and convex hulls over fields of Puiseux fractions

Moment Relaxations of Optimal Power Flow Problems: Beyond the Convex Hull

Optimal power flow (OPF) is one of the key electric power system optimization problems. "Moment" relaxations from the Lasserre hierarchy for polynomial optimization globally solve many OPF problems. Previous work illustrates the ability of higher-order moment relaxations to approach the convex hulls of OPF problems' non-convex feasible spaces. Using a small test case, this paper focuses on the ability of the moment relaxations to globally solve problems with objective functions that have unconstrained minima at infeasible points inside the convex hull of the non-convex constraints.

Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints

Let V be a semialgebraic set parameterized by quadratic polynomials over a quadratic set T. This paper studies semidefinite representation of its convex hull by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by a single quadratic constraint, we prove that its convex hull is equal to the first order moment type semidefinite relaxation of $V$, up to taking closures. Similar results hold when every quadratic polynomial is homogeneous and T is defined by two homogeneous quadratic constraints,or V is defined by rational quadratic parameterizations.

Convex Hull, IP and European Electricity Pricing in a European Power Exchanges setting with efficient computation of Convex Hull Prices

This paper introduces a computationally efficient comparative approach to classical pricing rules for day-ahead electricity markets, namely Convex Hull Pricing, IP Pricing and European-like market rules, in a Power Exchange setting with non-convex demand bids. These demand bids can, for example, be useful to large industrial consumers, and extend demand block orders in use by European Power Exchanges. For this purpose, we show that Convex Hull Prices can be efficiently computed using continuous relaxations for bidding products involving start-up costs, minimum power output levels and ramp cons…

通过凸包方法求解牛顿最小阻力问题的新数值解

New numerical solutions to Newton's problem of least resistance via a convex hull approach

提出基于凸包方法的数值方法求解凸函数类牛顿最小阻力问题，通过利用解的对称及极值点曲线分布特性，得到高精度解。

Extended formulations for convex hulls of graphs of bilinear functions

We consider the problem of characterizing the convex hull of the graph of a bilinear function $f$ on the $n$-dimensional unit cube $[0,1]^n$. Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a systematic study of properties of $f$ that guarantee that certain classes of BQP facets are sufficient for an extended formulation. We use a modification of Zuckerberg's geometric method for proving convex hull characterizations [Geometric proofs for convex hull defining formulations, Oper…

Semidefinite descriptions of the convex hull of rotation matrices

We study the convex hull of $SO(n)$, thought of as the set of $n\times n$ orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of $SO(n)$ is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.

Set characterizations and convex extensions for geometric convex-hull proofs

In the present work, we consider Zuckerberg's method for geometric convex-hull proofs introduced in [Geometric proofs for convex hull defining formulations, Operations Research Letters 44(5), 625-629 (2016)]. It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg's proof techn…

A Convex Primal Formulation for Convex Hull Pricing

In certain electricity markets, because of non-convexities that arise from their operating characteristics, generators that follow the independent system operator's (ISO's) decisions may fail to recover their cost through sales of energy at locational marginal prices. Discriminatory side payments are made by the ISO to incentivize the compliance of generators. Convex hull pricing is a uniform pricing scheme that minimizes these side payments. The Lagrangian dual problem of the unit commitment problem has been used to determine convex hull prices. However, this approach is computationally expen…

On convex hulls of epigraphs of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. O…

Convex Hull Representations for Bounded Products of Variables

It is well known that the convex hull of $\{(x,y,xy)\}$, where $(x,y)$ is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti {\em et al.\,}(2010) and Miller {\em et al.\,}(2011) showed that if there are additional upper and/or lower bounds on the product $z=xy$, then the convex hull can be represented by adding an infinite family of inequalities, requiring a separation algorithm to implement. Nguyen {\em et al.\,}(2018) derived convex hulls with bounds on $z$ for the more general case of $z=x^{b_1} y^{b_2}$, where $b_1\ge 1$, $b_2\ge 1…

The convex hull of a quadratic constraint over a polytope

A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. Solving non-convex QCQP to global optimality is a well-known NP-hard problem and a traditional approach is to use convex relaxations and branch-and-bound algorithms. This paper makes a contribution in this direction by showing that the exact convex hull of a general quadratic equation intersected with any bounded polyhedron is second-order cone representable. We present a simple constructive proof of…

关于简单单项式的图的凸包

On the convex hull of the graph of a simple monomial

本文研究了非负盒域中任意维度下简单单项式图的凸包，给出了线性不等式描述和体积公式。

A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation

In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we characterize the structure of the convex hull, and show that it can be formally studied using polyhedral theory. Moreover, we show that although perspective relaxation in the literature for this set fails to match the struct…

Explicit convex hull description of bivariate quadratic sets with indicator variables

We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $\mathcal S_2$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $\mathcal S_2$, w…

Convex Hull Formulations for Mixed-Integer Multilinear Functions

In this paper, we present convex hull formulations for a mixed-integer, multilinear term/function (MIMF) that features products of multiple continuous and binary variables. We develop two equivalent convex relaxations of an MIMF and study their polyhedral properties in their corresponding higher-dimensional spaces. We numerically observe that the proposed formulations consistently perform better than state-of-the-art relaxation approaches.

Spectrahedra and Convex Hulls of Rank-One Elements

The Helton-Nie Conjecture (HNC) is the proposition that every convex semialgebraic set is a spectrahedral shadow. Here we prove that HNC is equivalent to another propo- sition related to quadratically constrained quadratic programming. Namely, that the convex hull of the rank-one elements of any spectrahedron is a spectrahedral shadow. In the case of compact convex semialgebraic sets, the spectrahedra may be taken to be compact. We illustrate the relationship between spetrahedra and these convex subsets with examples.

Quadratic Optimization with Switching Variables: The Convex Hull for $n = 2$

We consider quadratic optimization in variables $(x,y)$ where $0\le x\le y$, and $y\in\{0,1\}^n$. Such binary $y$ are commonly refered to as "indicator" or "switching" variables and occur commonly in applications. One approach to such problems is based on representing or approximating the convex hull of the set $\{ (x,xx^T, yy^T) : 0\le x\le y\in\{0,1\}^n\}$. A representation for the case $n=1$ is known and has been widely used. We give an exact representation for the case $n=2$ by starting with a disjunctive representation for the convex hull and then eliminating auxilliary variables and cons…

二次不等式聚合与隐藏超平面凸性

Aggregations of quadratic inequalities and hidden hyperplane convexity

本文研究二次不等式定义集合的凸包，提出隐藏超平面凸性（HHC）概念，证明当相关二次映射满足HHC时凸包可由特殊聚合定义，并推广了聚合定义凸包的已知结果。

Modified convex hull pricing for power markets with price-sensitive load

We consider a general power market with price-sensitive consumer bids and non-convexities originating from supply (start-up and no-load costs, nonzero minimum output limits of generating units, etc.) and demand. The convex hull (minimum-uplift) pricing method produces the set of power prices that minimizes the total uplift payments to the market players needed to compensate their potential profits lost by accepting the centralized dispatch solution. All opportunities to supply (consume) any other output (consumption) volumes allowed by market player individual operational constraints are consi…

Micro Water-Energy Nexus: Optimal Demand-Side Management and Quasi-Convex Hull Relaxation

This paper investigates the water network's potential ability to provide demand response services to the power grid under the framework of a distribution-level water-energy nexus (micro-WEN). In particular, the hidden controllability of water loads, such as irrigation systems, was closely studied to improve the flexibility of electrical grids. A optimization model is developed for the demand-side management (DSM) of micro-WEN, and the simulation results assert that grid flexibility indeed benefits from controllable water loads. Although the proposed optimal DSM model is an intractable mixed-in…

A Note on the Convex Hull of Finitely Many Projections of Spectrahedra

A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of a spectrahedron. This improves upon the result of Helton and Nie, who prove the same result in the case of bounded sets.

从凸包到凹面：点集周围的曲面收缩

Form Convex Hull to Concavity: Surface Contraction Around a Point Set

本文针对d维点云凸包转凹面的问题，提出面片替换与扩张的迭代方法，构建能包裹所有点且贴合点云复杂几何的闭合无自交凹面Scc。

Stratifying High Dimensional Data Based on Proximity to the Convex Hull Boundary

The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine. One solution is to expand the list of high interest candidates to points lying near the boundary of the convex hull. We propose a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull. For each data point, a quadratic program is solved to determine an associated we…

On the convex hull of convex quadratic optimization problems with indicators

We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality descri…

Market Power in Convex Hull Pricing

The start up costs in many kinds of generators lead to complex cost structures, which in turn yield severe market loopholes in the locational marginal price (LMP) scheme. Convex hull pricing (a.k.a. extended LMP) is proposed to improve the market efficiency by providing the minimal uplift payment to the generators. In this letter, we consider a stylized model where all generators share the same generation capacity. We analyze the generators' possible strategic behaviors in such a setting, and then propose an index for market power quantification in the convex hull pricing schemes.

凸包定价的共正性特征

Copositive Characterizations of Convex Hull Pricing

针对机组组合非凸二元约束下无统一线性定价方案支持最优调度的问题，定义集中凸包价格，证明其与边际共正对偶价格在非退化时一致，并通过数值实验验证等价性及量化定价差距。

可达集的凸包

Convex Hulls of Reachable Sets

研究非线性系统在有界扰动和不确定初始条件下的可达集的凸包，提出基于常微分方程解的有限维表征，设计高效采样算法并分析误差界，应用于神经反馈回路分析和鲁棒MPC。

有理簇凸包的半定表示

Semidefinite representation of convex hulls of rational varieties

利用正半定矩矩阵和多项式平方和分解的对偶性质，证明了有理参数化代数簇的凸包在曲线、二次参数化超曲面和双变量四次参数化超曲面情形下具有半定表示。

A new proof of the description of the convex hull of space curves with totally positive torsion

We give new proofs of the description convex hulls of space curves $γ: [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(γ', γ'', \ldots, γ^{(d)})$ are positive. In particular, we recover parametric representation of the boundary of the convex hull, different formulas for its surface area and the volume of the convex hull, and the solution to a general moment problem corresponding to $γ$.