Univariate Polynomial Optimization with Sum-of-Squares Interpolants

2017 ◽  
pp. 143-162
Author(s):  
Dávid Papp
Keyword(s):  
Polynomial Optimization ◽  
Sum Of Squares ◽  
Univariate Polynomial

Sparse Sum-of-Squares Optimization for Model Updating Through Minimization of Modal Dynamic Residuals

2019 ◽  
Vol 2 (1) ◽  
Author(s):  
Dan Li ◽  
Yang Wang
Keyword(s):  
Optimization Problem ◽  
Model Updating ◽  
Polynomial Optimization ◽  
Element Model ◽  
Optimization Method ◽  
Sum Of Squares ◽  
Global Optimality ◽  
Polynomial Optimization Problem ◽  
Computation Efficiency ◽  
Dynamic Residuals

This research investigates the application of sum-of-squares (SOS) optimization method on finite element model updating through minimization of modal dynamic residuals. The modal dynamic residual formulation usually leads to a nonconvex polynomial optimization problem, the global optimality of which cannot be guaranteed by most off-the-shelf optimization solvers. The SOS optimization method can recast a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem. However, the size of the SDP problem can grow very large, sometimes with hundreds of thousands of variables. To improve the computation efficiency, this study exploits the sparsity in SOS optimization to significantly reduce the size of the SDP problem. A numerical example is provided to validate the proposed method.

scholarly journals On the Construction of Converging Hierarchies for Polynomial Optimization Based on Certificates of Global Positivity

2019 ◽  
Vol 44 (4) ◽  
pp. 1192-1207 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Georgina Hall
Keyword(s):  
20Th Century ◽  
Lower Bounds ◽  
Polynomial Optimization ◽  
Sum Of Squares ◽  
Late 20Th Century ◽  
Minimization Problems ◽  
General Polynomial ◽  
Feasible Sets ◽  
Diagonally Dominant ◽  
Inner Approximation

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellensätze from the late 20th century (e.g., due to Stengle, Putinar, or Schmüdgen) that certify positivity of a polynomial on an arbitrary closed basic semialgebraic set. In this paper, we show that such hierarchies could in fact be designed from much more limited Positivstellensätze dating back to the early 20th century that only certify positivity of a polynomial globally. More precisely, we show that any inner approximation to the cone of positive homogeneous polynomials that is arbitrarily tight can be turned into a converging hierarchy of lower bounds for general polynomial minimization problems with compact feasible sets. This in particular leads to a semidefinite programming–based hierarchy that relies solely on Artin’s solution to Hilbert’s 17th problem. We also use a classical result from Pólya on global positivity of even forms to construct an “optimization-free” converging hierarchy for general polynomial minimization problems with compact feasible sets. This hierarchy requires only polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming–based and second-order cone programming–based hierarchies for polynomial minimization problems that rely on the recently introduced concepts of diagonally dominant sum of squares and scaled diagonally dominant sum of squares polynomials. We remark that the scope of this paper is theoretical at this stage, as our hierarchies—though they involve at most two sum of squares constraints or only elementary arithmetic at each level—require the use of bisection and increase the number of variables (respectively, the degree) of the problem by the number of inequality constraints plus three (respectively, by a factor of two).

PEMODELAN MATEMATIS ADSORPSI CO2 DENGAN STRONG BASE ANION EXCHANGE RESIN SEBAGAI ADSORBEN UNTUK PURIFIKASI BIOGAS

2015 ◽  
Vol 5 (1) ◽  
pp. 11
Author(s):  
Anies Mutiari ◽  
Wiratni Wiratni ◽  
Aswati Mindaryani
Keyword(s):  
Anion Exchange ◽  
Pilot Plant ◽  
Exchange Resin ◽  
Anion Exchange Resin ◽  
Scale Up ◽  
Sum Of Squares ◽  
Strong Base ◽  
Model Transfer ◽  
Plant Parameter

Pemurnian biogas telah banyak dilakukan untuk menghilangkan kadar CO2  dan meningkatkan kandungan CH4  yang terkandung di dalamnya. Kandungan CH4 yang tinggi akan memberikan unjuk kerja yang lebih baik. Model  matematis proses adsorpsi CO2 disusun berdasarkan teori lapisan film antar fasa, dimana pada proses yang ditinjau terdapat tiga fase yaitu gas, cair dan padat. Model matematis dari data eksperimental   kecepatan dan kesetimbangan proses adsorpsi CO2 melalui mekanisme pertukaran ion di suatu kolom adsorpsi telah dibuat. Model ini dibuat untuk mencari konstanta yang dapat dipergunakan pada proses scale up data laboratorium ke skala pilot plant. Parameter proses kecepatan yang dicari nilainya adalah koefisien transfer massa massa volumetris CO2 pada fase cair (kLa), koefisien transfer massa volumetris CO2 pada fasegas (kGa) dan tetapan laju reaksi (k1 dan k2). Pada hasil penelitian ini ditunjukkan bahwa nilai parameter yang diperoleh sesuai hasil fitting data dengan model matematis yang digunakan, yaitu model transfer massa pada lapisan film antar fase secara seri: adalah kGa, kla, k1 dan k2  dengan nilai Sum of Squares Error (SSE) rata-rata 0,0431. Perbandingan nilai kGa hasil simulasi dan teoritisnya memberikan kesalahan rata-rata 18,79%. Perbandingan nilai kLa hasil simulasi dan teoritis memberikan kesalahan rata-rata 7,92%.Kata kunci: model matematis, adsorpsi CO2, pemurnian biogas

On the Representation of a Number as a Sum of Squares

1895 ◽  
Vol s1-27 (1) ◽  
pp. 55-60 ◽  
Author(s):  
G. B. Mathews
Keyword(s):  
Sum Of Squares

scholarly journals Sum-of-Squares based computation of a Lyapunov function for proving stability of a satellite with electromagnetic actuation

2020 ◽  
Vol 53 (2) ◽  
pp. 7380-7385
Author(s):  
Rahul Misra ◽  
Rafał Wisniewski ◽  
Özkan Karabacak
Keyword(s):  
Lyapunov Function ◽  
Sum Of Squares ◽  
Electromagnetic Actuation
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