quinopt()

Minimize a cost function subject to quadratic integral inequality constraints, or reset QUINOPT.

Syntax:quinopt clear or quinopt('clear')
Description:clears QUINOPT’s internal variables, to be used in combination with MATLAB’s clear command.
Syntax:quinopt(EXPR)
Description:tests whether a quadratic integral inequality with integrand specified by EXPR is feasible using a finite dimensional relaxation based on semidefinite programming. If multiple integral inequalities must be tested simultaneously, EXPR can be a vector such that the i-th entry specifies the integrand of the i-th integral inequality. Each entry of EXPR must be a polynomial of the integration variable returned by the command indvar(), and a quadratic polynomial of the dependent variables returned by the function depvar().
Syntax:quinopt(EXPR,BC)
Description:determines whether the integral inequalities, with integrand specified by EXPR, are feasible for all dependent variables satisfying the homogeneous boundary conditions specified by the vector BC. Specifically, BC is interpreted as the list of boundary conditions BC(1)=0, …, BC(end)=0. Like EXPR, BC must be created using the variables returned by the commands indvar() and depvar().
Syntax:quinopt(EXPR,BC,OBJ)
Description:minimizes the objective function OBJ constrained by the integral inequalities specified by EXPR and BC.
Syntax:

quinopt(EXPR,BC,OBJ,OPTIONS)
Description:

overrides the default options. OPTIONS is a structure containing any of the following fields:

  • OPTIONS.YALMIP: a substructure containing the options for YALMIP, set with YALMIP’s command sdpsettings().
  • OPTIONS.N: an integer specifying the number of Legendre coefficients to use in the expansion of the dependent variable to obtain an SDP-representable relaxation of the quadratic integral inequality.
  • OPTIONS.method: if set to 'inner' (default), QUINOPT generates an inner approximation of the feasible set of the integral inequalities specified by EXPR, i.e. the quadratic integral inequality is strenghtened. If set to 'outer', an outer approximation is constructed, i.e. the integral inequality is relaxed into a weaker constraint.
  • OPTIONS.BCprojectorBasis: string specifying which basis to use for the projection on the boundary conditions. If set to 'rref' (default), use a “rational” basis. If set to 'orth', use an orthonormal basis. The orthonormal basis may be preferable numerically, but it may destroy sparsity of the data.
  • OPTIONS.sosdeg: the degree of the sum-of-squares polynomials used in the S-procedure to localize SOS constraints from the integral inequality to the integration domain. Default value: 6.
  • OPTIONS.solve: if set to 'true' (default), QUINOPT calls the solver specified by the YALMIP options (or YALMIP’s default solver). If set to 'false', QUINOPT does not call the solver, but simply sets up the YALMIP problem structure. In this case, additional outputs to QUINOPT are required (see below).
Syntax:quinopt(EXPR,BC,OBJ,OPTIONS,CNSTR) or quinopt(EXPR,BC,OBJ,OPTIONS,CNSTR,PARAMETERS)
Description:minimizes the objective function OBJ subjet to the integral inequalities specified by EXPR and BC, and the additional constraints given by CNSTR. CNSTR is a constraint object built with YALMIP. If CNSTR contains sum-of-square constraints, then the variable parameters in the polynomial expressions must be specified in the input vector PARAMETERS. See YALMIP’s function sos() and solvesos() for more details on specifying sum-of-squares constraints with YALMIP.
Syntax:

[SOL,CNSTR,DATA] = quinopt(...)
Description:

returns solution informations in the structure SOL, the set of YALMIP constraint CNSTR used to solve the optimization problem, and a structure DATA containing all variables used to set up the constraints in CNSTR. The solution structure SOL contains the following fields:

  • SOL.setupTime: the time taken to set up the problem

  • SOL.solutionTime: the time taken by YALMIP to solve the problem

  • SOL.problem: code of problem encountered during setup. Values are:

    • 0: no problem
    • 1: ill-posed inequality
    • 2: infeasible relaxation

  • SOL.FeasCode: code for the feasibility of the solution returned by YALMIP. Run quinoptFeasCode at the MATLAB command line to obtain a complete list.

  • SOL.YALMIP: the solution structure returned by YALMIP. See YALMIP’s functions optimize() and solvesos() for more details.