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## Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making

### Citations

1996 | Nonlinear Systems - Khalil - 1992 |

1098 | Semidefinite programming
- Vandenberghe, Boyd
- 1996
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Citation Context ...etric n×n matrix A is positive semidefinite (psd) if xTAx ≥ 0,∀x ∈ Rn, and that semidefinite programming is the problem of optimizing over psd matrices subject to affine inequalities on their entries =-=[43]-=-. We denote the positive semidefiniteness of a matrix A with the standard notation A 0. Theorem 2.1 (see, e.g., [35],[36]). A multivariate polynomial p(x) in n variables and of degree 2d is a sum of... |

456 |
On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents
- Dubins
- 1957
(Show Context)
Citation Context ...asoning about families of trajectories that the system could end up following, making the problem more challenging. The states and dynamics of the UAV are inspired by the widely-used Dubins car model =-=[18]-=- and are given by: x = xy ψ , ẋ = f(x,u,w) = ẋẏ ψ̇ = −v sinψ + wv cosψ u , (7) where x and y are the x and y positions of the UAV in the environment, v = 1 m/s is the speed of the... |

407 |
Optimal Control Linear Quadratic Methods
- Anderson, Moore
- 1990
(Show Context)
Citation Context ...he iteration is then a SDSOS program. This iterative procedure is described in more detail in [25] and can be initialized with the Lyapunov function from a Linear Quadratic Regulator (LQR) controller =-=[7]-=-. The iterations are terminated when the objective changes by less than 1 percent. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x (m) ys(m ) (a) x – y subspace... |

384 | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization
- Parrilo
- 2000
(Show Context)
Citation Context ...m of optimizing over psd matrices subject to affine inequalities on their entries [43]. We denote the positive semidefiniteness of a matrix A with the standard notation A 0. Theorem 2.1 (see, e.g., =-=[35]-=-,[36]). A multivariate polynomial p(x) in n variables and of degree 2d is a sum of squares if and only if there exists a symmetric matrix Q (often called the Gram matrix) such that p(x) = zTQz, Q 0,... |

361 | Semidefinite programming relaxations for semialgebraic problems
- Parrilo
(Show Context)
Citation Context ...a basic semialgebraic set using sos polynomials. These certificates are only slightly more complicated than (1) and involve sos multipliers associated with products among polynomials gi that define S =-=[36]-=-. A great reference for the interested reader is the survey paper by Laurent [24]. The computational advantage of a certificate of (global or local) nonnegativity via sum of squares polynomials is tha... |

270 |
Positive polynomials on compact semi-algebraic sets
- Putinar
- 1993
(Show Context)
Citation Context ...eed, if we evaluate the above expression at any x ∈ S, nonnegativity of the polynomials s0, s1 . . . , sm imply that p(x) ≥ 0. A Positivstellensatz theorem from real algebraic geometry due to Putinar =-=[40]-=- states that if the set S satisfies the so-called Archimedean property, a property only slightly stronger than compactness2, then every polynomial positive on S has a representation of the type (1), f... |

244 | Second-order cone programming. - Alizadeh, Goldfarb - 2003 |

178 |
Dynamic Programming and Optimal Control, Volume 1
- Bertsekas
- 1995
(Show Context)
Citation Context ...ansportation engineering. While techniques in more established areas of optimization theory such as linear, integer, combinatorial, and dynamic programming have found wide applications in these areas =-=[11, 10, 9, 22]-=-, the relatively newer field of polynomial optimization, which has gone through rapid advancements in recent years, may yet prove to reveal many unexplored applications. It is our aim in this paper to... |

176 |
Introduction to Linear Optimization
- Bertsimas, Tsitsiklis
- 1997
(Show Context)
Citation Context ...ansportation engineering. While techniques in more established areas of optimization theory such as linear, integer, combinatorial, and dynamic programming have found wide applications in these areas =-=[11, 10, 9, 22]-=-, the relatively newer field of polynomial optimization, which has gone through rapid advancements in recent years, may yet prove to reveal many unexplored applications. It is our aim in this paper to... |

154 |
Some NP-complete problems in quadratic and nonlinear programming
- Murty, Kabadi
- 1987
(Show Context)
Citation Context ...a convex optimization The task of optimizing over nonnegative polynomials or even checking nonnegativity of a given polynomial, either globally or on a basic semialgebraic set, is known to be NP-hard =-=[33]-=-. This is true already for checking global nonnegativity of a quartic (degree-4) polynomial, or for checking nonnegativity of a quadratic polynomial on a set defined by linear inequalities. A popular ... |

152 | Sums of squares, moment matrices and optimization over polynomials, IMA Volume Emerging Applications of Algebraic Geometry
- Laurent
- 2009
(Show Context)
Citation Context ...htly more complicated than (1) and involve sos multipliers associated with products among polynomials gi that define S [36]. A great reference for the interested reader is the survey paper by Laurent =-=[24]-=-. The computational advantage of a certificate of (global or local) nonnegativity via sum of squares polynomials is that it can be automatically found by semidefinite programming. What establishes the... |

126 |
A Nullstellensatz and a Positivstellensatz in semialgebraic geometry
- Stengle
- 1974
(Show Context)
Citation Context ...s s0, s1, . . . , sm of high enough degree (see also [34] for degree bounds). Even with absolutely no qualifications about the set S, there are other Positivstellensatz theorems (e.g., due to Stengle =-=[41]-=-) that certify nonnegativity of a polynomial on a basic semialgebraic set using sos polynomials. These certificates are only slightly more complicated than (1) and involve sos multipliers associated w... |

116 | Quadrotor helicopter flight dynamics and control: Theory and experiment,”
- Hoffmann, Huang, et al.
- 2007
(Show Context)
Citation Context ... Quadrotor Model Quadrotors (see Figure 7) have recently been recognized as a popular platform for academic research in systems theory due to their agile maneuvering capabilities and inexpensive cost =-=[31, 20]-=-. They have also been considered for the task of load transportation, not only in laboratory settings [32]5, but also by the aerospace companies Bell and Boeing and the online retail company Amazon6. ... |

101 |
Über die Darstellung definiter Formen als Summe von Formenquadraten,
- Hilbert
- 1965
(Show Context)
Citation Context ... We say that a polynomial p is a sum of squares (sos), if it can be written as p = ∑ i q 2 for some other polynomials qi. Obviously, such a decomposition is a sufficient (but in general not necessary =-=[19]-=-) condition for (global) nonnegativity of p. The situation where p is only constrained to be nonnegative on a certain basic semialgebraic set1 S := {x ∈ Rn| g1(x) ≥ 0, . . . , gm(x) ≥ 0} 1In this form... |

89 | Safety verification of hybrid systems using barrier certificates
- Prajna, Jadbabaie
- 2004
(Show Context)
Citation Context ...fe and we want to guarantee that trajectories starting in Ssafe would never end up in Sunsafe. This guarantee can be achieved if we succeed in finding a function V : Rn → R, called a barrier function =-=[38]-=-, [39], [8], with the following three properties: V (x) < 1 ∀x ∈ Ssafe, V > 1 ∀x ∈ Sunsafe, V̇ (x) ≤ 0 ∀x. The expression V̇ denotes the time derivative of V along trajectories. If V is a polynomial, ... |

82 |
Trajectory generation and control for precise aggressive maneuvers with quadrotors,”
- Mellinger, Michael, et al.
- 2012
(Show Context)
Citation Context ...of attraction (ROA), i.e., the set of initial conditions the controller is guaranteed to stabilize to the goal position. Figure 7: We design a hovering controller for the quadrotor model described in =-=[30]-=-. (Image from [30].) We use the dynamics model described in [30] for our numerical experiments. The model includes 16 5A video corresponding to the paper is available at https://www.youtube.com/watch?... |

63 | A Servey of S-lemma
- Pólik, Terlaky
(Show Context)
Citation Context ... (4) is quadratic. If a quadratic polynomial is nonnegative on a region defined by another quadratic, this fact is always certified by a constant degree multiplier—this is the celebrated S-lemma; see =-=[37]-=-. Similarly, if we solve the problem for the transmitter located at (1, 1.5), the optimal value of (5) which matches the optimal value of (3) is 11.446. So our task is indeed not achievable with one t... |

61 |
CVXGEN: A code generator for embedded convex optimization
- Mattingley, Boyd
(Show Context)
Citation Context ...mentation of this approach on a hardware platform is plausible. Such a hardware implementation can benefit from already-existing SOCP solvers that are specifically designed to run on embedded systems =-=[29]-=-, [28]. In particular, [17] presents an approach for generating stand-alone C code for an SOCP solver that can run very efficiently and with low memory footprint. The use of such real-time SOCP solver... |

51 | A framework for worst-case and stochastic safety verification using barrier certificates
- Prajna, Jadbabaie, et al.
(Show Context)
Citation Context ... we want to guarantee that trajectories starting in Ssafe would never end up in Sunsafe. This guarantee can be achieved if we succeed in finding a function V : Rn → R, called a barrier function [38], =-=[39]-=-, [8], with the following three properties: V (x) < 1 ∀x ∈ Ssafe, V > 1 ∀x ∈ Sunsafe, V̇ (x) ≤ 0 ∀x. The expression V̇ denotes the time derivative of V along trajectories. If V is a polynomial, V̇ wil... |

39 | On the complexity of Putinar’s Positivstellensatz,
- Nie, Schweighofer
- 2007
(Show Context)
Citation Context ...roperty only slightly stronger than compactness2, then every polynomial positive on S has a representation of the type (1), for some sos polynomials s0, s1, . . . , sm of high enough degree (see also =-=[34]-=- for degree bounds). Even with absolutely no qualifications about the set S, there are other Positivstellensatz theorems (e.g., due to Stengle [41]) that certify nonnegativity of a polynomial on a bas... |

37 | Real-time convex optimization in signal processing
- Mattingley, Boyd
- 2010
(Show Context)
Citation Context ...ion of this approach on a hardware platform is plausible. Such a hardware implementation can benefit from already-existing SOCP solvers that are specifically designed to run on embedded systems [29], =-=[28]-=-. In particular, [17] presents an approach for generating stand-alone C code for an SOCP solver that can run very efficiently and with low memory footprint. The use of such real-time SOCP solvers has ... |

33 | Cooperative grasping and transport using multiple quadrotors. In:
- Mellinger, Shomin, et al.
- 2010
(Show Context)
Citation Context ...search in systems theory due to their agile maneuvering capabilities and inexpensive cost [31, 20]. They have also been considered for the task of load transportation, not only in laboratory settings =-=[32]-=-5, but also by the aerospace companies Bell and Boeing and the online retail company Amazon6. In this section, we consider the problem of designing a nonlinear stabilizing feedback controller for the ... |

30 | Some controls applications of sum of squares programming
- Jarvis-Wloszek, Feeley, et al.
- 2003
(Show Context)
Citation Context ...problem (12) is not convex in general since it involves conditions that are bilinear in the decision variables. However, problems of this nature are common in the SOS programming literature (see e.g. =-=[21]-=-) and are typically solved by iteratively optimizing groups of decision variables. Each step in the iteration is then a SDSOS program. This iterative procedure is described in more detail in [25] and ... |

14 | The wireless network jamming problem
- Commander, Pardalos, et al.
- 2007
(Show Context)
Citation Context ...eographical domains where a wireless service provider would like to guarantee a certain level of signal quality. The problem we describe is motivated by some interesting and relatively recent work in =-=[16]-=-, [15] (see also the thesis [14]), where the motivation is instead to jam the communication network of an adversary with a wireless transmitter. We note, however, that there are a few differences betw... |

13 | Control design along trajectories with sums of squares programming
- Majumdar, Ahmadi, et al.
- 2013
(Show Context)
Citation Context ... catch up to the desired speed). In the end our system takes the form ẋ = f(x) + g(x)u(x) with f and g given and the control u as a decision function. We use the method presented in our earlier work =-=[25]-=- in collaboration with Russ Tedrake to design a hovering controller u for the system. The fixed point corresponding to the hovering configuration has the first twelve states of the system equaling 0 b... |

13 | Robust online motion planning with regions of finite time invariance
- Majumdar, Tedrake
- 2012
(Show Context)
Citation Context ...cular set of obstacles (but does not consider the case where obstacle positions are not known beforehand and decisions must be made in real time). Similarly, other previous SOS programming approaches =-=[27]-=- to collision avoidance have involved solving SOS programs offline and then using these precomputed results to do planning in real time. In contrast, in our example here, the optimization problems are... |

11 | ECOS: An SOCP solver for embedded systems.
- Domahidi, Chu, et al.
- 2013
(Show Context)
Citation Context ...on a hardware platform is plausible. Such a hardware implementation can benefit from already-existing SOCP solvers that are specifically designed to run on embedded systems [29], [28]. In particular, =-=[17]-=- presents an approach for generating stand-alone C code for an SOCP solver that can run very efficiently and with low memory footprint. The use of such real-time SOCP solvers has already been consider... |

9 | Algebraic relaxations and hardness results in polynomial optimization and Lyapunov analysis - Ahmadi - 2011 |

8 |
Location-Routing Problems with Uncertainty. Facilities Location
- Berman, Jaillet, et al.
- 1995
(Show Context)
Citation Context ...ansportation engineering. While techniques in more established areas of optimization theory such as linear, integer, combinatorial, and dynamic programming have found wide applications in these areas =-=[11, 10, 9, 22]-=-, the relatively newer field of polynomial optimization, which has gone through rapid advancements in recent years, may yet prove to reveal many unexplored applications. It is our aim in this paper to... |

8 |
Integer programming: theory and practice:
- Karlof
- 2005
(Show Context)
Citation Context |

6 |
Minimum landing error powered descent guidance for mars landing using convex optimization
- Blackmore, Açıkmeşe, et al.
(Show Context)
Citation Context ... very efficiently and with low memory footprint. The use of such real-time SOCP solvers has already been considered for tasks such as landing of spacecraft (e.g., for NASA’s Mars exploration project) =-=[12]-=-. A particular example of a barrier function computed for the controller u1 is shown in Figure 4. The obstacles are shown in red and the initial state of the UAV is also plotted. The 1-level set of th... |

5 |
On factor width and symmetric H-matrices, Linear Algebra and its
- Boman, Chen, et al.
- 2005
(Show Context)
Citation Context ...AD (or equivalently, DAD) is diagonally dominant. The set of n× n sdd matrices will be denoted by SDDn. We note that sdd matrices are sometimes referred to as generalized diagonally dominant matrices =-=[13]-=-. Theorem 2.4. The set of matrices SDDn can be optimized over using second order cone programming. Proof. Positive semidefiniteness of the 2 × 2 matrices in Definition 2.3 is equivalent to the diagona... |

5 | Jamming communication networks under complete uncertainty
- Commander, Pardalos, et al.
(Show Context)
Citation Context ...hical domains where a wireless service provider would like to guarantee a certain level of signal quality. The problem we describe is motivated by some interesting and relatively recent work in [16], =-=[15]-=- (see also the thesis [14]), where the motivation is instead to jam the communication network of an adversary with a wireless transmitter. We note, however, that there are a few differences between ou... |

4 | Safety verification of reactive controllers for UAV flight in cluttered environments using barrier certificates
- Barry, Majumdar, et al.
- 2012
(Show Context)
Citation Context ...nt to guarantee that trajectories starting in Ssafe would never end up in Sunsafe. This guarantee can be achieved if we succeed in finding a function V : Rn → R, called a barrier function [38], [39], =-=[8]-=-, with the following three properties: V (x) < 1 ∀x ∈ Ssafe, V > 1 ∀x ∈ Sunsafe, V̇ (x) ≤ 0 ∀x. The expression V̇ denotes the time derivative of V along trajectories. If V is a polynomial, V̇ will als... |

4 |
Optimization problems in telecommunications with military applications
- Commander
- 2007
(Show Context)
Citation Context ...less service provider would like to guarantee a certain level of signal quality. The problem we describe is motivated by some interesting and relatively recent work in [16], [15] (see also the thesis =-=[14]-=-), where the motivation is instead to jam the communication network of an adversary with a wireless transmitter. We note, however, that there are a few differences between our setting and that of [16]... |

4 |
Recent advances in quadrotor capabilities,”
- Mellinger, Michael, et al.
- 2011
(Show Context)
Citation Context ... Quadrotor Model Quadrotors (see Figure 7) have recently been recognized as a popular platform for academic research in systems theory due to their agile maneuvering capabilities and inexpensive cost =-=[31, 20]-=-. They have also been considered for the task of load transportation, not only in laboratory settings [32]5, but also by the aerospace companies Bell and Boeing and the online retail company Amazon6. ... |

2 |
DSOS and SDSOS optimization: More tractable alternatives to SOS optimization
- Ahmadi, Majumdar
- 2014
(Show Context)
Citation Context ...opular approach for certifying polynomial nonnegativity. While remarkably powerful, it often faces scalability limitations on larger-scale problems. As a potential remedy, we have recently introduced =-=[5, 4]-=- the concepts of diagonally dominant and scaled diagonally dominant sum of squares (dsos and sdsos) decomposition, which instead of SDP result in linear programs (LP) and second order cone programs (S... |

2 |
SeDuMi version 1.05, Oct. 2001. Latest version available at http://sedumi.ie.lehigh.edu
- Sturm
(Show Context)
Citation Context ... 16 variables and degree 6 and result in a Gram matrices with about half a million decision variables. A semidefinite constraint of this size is quite expensive—for example, the SDP solvers of SeDuMi =-=[42]-=- and MOSEK [2] fail to solve the quadrotor problem on our machine and quickly run out of memory. 2.1 DSOS and SDSOS Optimization In order to address the problem of scalability posed by SDP, we have re... |

1 |
DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization
- Ahmadi, Majumdar
- 2014
(Show Context)
Citation Context ...opular approach for certifying polynomial nonnegativity. While remarkably powerful, it often faces scalability limitations on larger-scale problems. As a potential remedy, we have recently introduced =-=[5, 4]-=- the concepts of diagonally dominant and scaled diagonally dominant sum of squares (dsos and sdsos) decomposition, which instead of SDP result in linear programs (LP) and second order cone programs (S... |

1 | Control and verification of high-dimensional systems via DSOS and SDSOS optimization
- Majumdar, Ahmadi, et al.
- 2014
(Show Context)
Citation Context ... may choose one approach over the other. In this paper, we will be using SOS optimization (Section 3) and SDSOS optimization (Sections 5 and 6) in our numerical experiments. The reader is referred to =-=[26, 4, 5]-=- for many numerical examples involving DSOS optimization. We also remark in passing that SDSOS or even DSOS programming enjoy many of the same theoretical (asymptotic) guarantees of SOS programming—re... |