|
|
#pragma once
|
|
|
#include "drake/solvers/mathematical_program.h"
|
|
|
#include "drake/solvers/mathematical_program_result.h"
|
|
|
|
|
|
namespace drake {
|
|
|
namespace solvers {
|
|
|
/**
|
|
|
* This is example 3.35 from Semidefinite Optimization and Convex Algebraic
|
|
|
* Geometry by G. Blekherman, P. Parrilo and R. Thomas. Solve a semidefinite
|
|
|
* programming problem to verify that the univariate quartic polynomial p(x)
|
|
|
* = x⁴+4x³+6x²+4x+5 is sum-of-squares (sos).
|
|
|
*/
|
|
|
class UnivariateQuarticSos {
|
|
|
public:
|
|
|
DRAKE_NO_COPY_NO_MOVE_NO_ASSIGN(UnivariateQuarticSos)
|
|
|
|
|
|
UnivariateQuarticSos();
|
|
|
|
|
|
const MathematicalProgram& prog() const { return prog_; }
|
|
|
|
|
|
void CheckResult(const MathematicalProgramResult& result, double tol) const;
|
|
|
|
|
|
private:
|
|
|
MathematicalProgram prog_;
|
|
|
symbolic::Polynomial p_;
|
|
|
MatrixXDecisionVariable gram_;
|
|
|
VectorX<symbolic::Monomial> monomial_basis_;
|
|
|
};
|
|
|
|
|
|
/**
|
|
|
* This is example 3.38 from Semidefinite Optimization and Convex Algebraic
|
|
|
* Geometry by G. Blekherman, P. Parrilo and R. Thomas. Solve a semidefinite
|
|
|
* programming problem to verify that the bivariate quartic polynomial p(x, y) =
|
|
|
* 2x⁴+5y⁴−2x²y²+2x³y+2x+2 is sum-of-squares (sos).
|
|
|
*/
|
|
|
class BivariateQuarticSos {
|
|
|
public:
|
|
|
DRAKE_NO_COPY_NO_MOVE_NO_ASSIGN(BivariateQuarticSos)
|
|
|
|
|
|
BivariateQuarticSos();
|
|
|
|
|
|
const MathematicalProgram& prog() const { return prog_; }
|
|
|
|
|
|
void CheckResult(const MathematicalProgramResult& result, double tol) const;
|
|
|
|
|
|
private:
|
|
|
MathematicalProgram prog_;
|
|
|
symbolic::Polynomial p_;
|
|
|
MatrixXDecisionVariable gram_;
|
|
|
VectorX<symbolic::Monomial> monomial_basis_;
|
|
|
};
|
|
|
|
|
|
/**
|
|
|
* This is example 3.50 from Semidefinite Optimization and Convex Algebraic
|
|
|
* Geometry by G. Blekherman, P. Parrilo and R. Thomas. Solve a simple sos
|
|
|
* program
|
|
|
* max a + b
|
|
|
* s.t x⁴ + ax + 2+b is sos
|
|
|
* (a-b+1)x² + bx + 1 is sos
|
|
|
*/
|
|
|
class SimpleSos1 {
|
|
|
public:
|
|
|
DRAKE_NO_COPY_NO_MOVE_NO_ASSIGN(SimpleSos1)
|
|
|
|
|
|
SimpleSos1();
|
|
|
|
|
|
const MathematicalProgram& prog() const { return prog_; }
|
|
|
|
|
|
void CheckResult(const MathematicalProgramResult& result, double tol) const;
|
|
|
|
|
|
private:
|
|
|
MathematicalProgram prog_;
|
|
|
symbolic::Variable a_;
|
|
|
symbolic::Variable b_;
|
|
|
symbolic::Variable x_;
|
|
|
symbolic::Polynomial p1_;
|
|
|
symbolic::Polynomial p2_;
|
|
|
MatrixXDecisionVariable gram1_;
|
|
|
MatrixXDecisionVariable gram2_;
|
|
|
VectorX<symbolic::Monomial> monomial_basis1_;
|
|
|
VectorX<symbolic::Monomial> monomial_basis2_;
|
|
|
};
|
|
|
|
|
|
/**
|
|
|
* Prove that the Motzkin polynomial m(x, y) = x⁴y² + x²y⁴ + 1 − 3x²y² is
|
|
|
* always non-negative.
|
|
|
* One ceritificate for the proof is the existence of a polynomial r(x, y)
|
|
|
* satisfying r(x, y) being sos and r(x, y) > 0 for all x, y ≠ 0, such that
|
|
|
* r(x, y) * m(x, y) is sos.
|
|
|
* So we solve the following problem
|
|
|
* find r(x, y)
|
|
|
* s.t r(x, y) - x² − y² is sos
|
|
|
* r(x, y) * m(x, y) is sos
|
|
|
*/
|
|
|
class MotzkinPolynomial {
|
|
|
public:
|
|
|
DRAKE_NO_COPY_NO_MOVE_NO_ASSIGN(MotzkinPolynomial)
|
|
|
|
|
|
MotzkinPolynomial();
|
|
|
|
|
|
const MathematicalProgram& prog() const { return prog_; }
|
|
|
|
|
|
void CheckResult(const MathematicalProgramResult& result, double tol) const;
|
|
|
|
|
|
private:
|
|
|
MathematicalProgram prog_;
|
|
|
symbolic::Variable x_;
|
|
|
symbolic::Variable y_;
|
|
|
symbolic::Polynomial m_;
|
|
|
symbolic::Polynomial r_;
|
|
|
MatrixXDecisionVariable gram1_;
|
|
|
MatrixXDecisionVariable gram2_;
|
|
|
VectorX<symbolic::Monomial> monomial_basis1_;
|
|
|
VectorX<symbolic::Monomial> monomial_basis2_;
|
|
|
};
|
|
|
|
|
|
/**
|
|
|
* Solve the following optimization problem for a univariate polynomial
|
|
|
* max a + b + c
|
|
|
* s.t p(x) = x⁴+ax³+bx²+c+1>=0 for all x >= 0
|
|
|
* p(1) = 1
|
|
|
* According to theorem 3.71 in Semidefinite Optimization and Convex Algebraic
|
|
|
* Geometry by G. Blekherman, P. Parrilo and R. Thomas, this is equivalent to
|
|
|
* the following SOS problem
|
|
|
* max a + b + c
|
|
|
* s.t p(x) = s(x) + x * t(x)
|
|
|
* p(1) = 1
|
|
|
* s(x) is sos
|
|
|
* t(x) is sos
|
|
|
*/
|
|
|
class UnivariateNonnegative1 {
|
|
|
public:
|
|
|
DRAKE_NO_COPY_NO_MOVE_NO_ASSIGN(UnivariateNonnegative1)
|
|
|
|
|
|
UnivariateNonnegative1();
|
|
|
|
|
|
const MathematicalProgram& prog() const { return prog_; }
|
|
|
|
|
|
void CheckResult(const MathematicalProgramResult& result, double tol) const;
|
|
|
|
|
|
private:
|
|
|
MathematicalProgram prog_;
|
|
|
symbolic::Variable a_;
|
|
|
symbolic::Variable b_;
|
|
|
symbolic::Variable c_;
|
|
|
symbolic::Variable x_;
|
|
|
symbolic::Polynomial p_;
|
|
|
symbolic::Polynomial s_;
|
|
|
symbolic::Polynomial t_;
|
|
|
MatrixXDecisionVariable gram_s_;
|
|
|
MatrixXDecisionVariable gram_t_;
|
|
|
};
|
|
|
} // namespace solvers
|
|
|
} // namespace drake
|
