Canonical HTML: https://autolyap.github.io/iteration_independent_analysis/ Documentation version: 0.2.1 # Iteration-independent analysis ```text class autolyap.IterationIndependent ``` Bases: `object` Iteration-independent Lyapunov analysis utilities. For the mathematical formulation, notation, and convergence statements, see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). This class provides the corresponding computational interface, with [`search_lyapunov()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.search_lyapunov "autolyap.IterationIndependent.search_lyapunov") as the main entry point. ```text static search_lyapunov( prob: InclusionProblem, algo: Algorithm, P: ndarray, T: ndarray, p: ndarray | None = None, t: ndarray | None = None, rho: float = 1.0, h: int = 0, alpha: int = 0, Q_equals_P: bool = False, S_equals_T: bool = False, q_equals_p: bool = False, s_equals_t: bool = False, remove_C2: bool = False, remove_C3: bool = False, remove_C4: bool = True, solver_options: SolverOptions | None = None, verbosity: int = 1, ) → Mapping[str, Any] ``` [InclusionProblem](https://autolyap.github.io/problem_class/#autolyap.problemclass.InclusionProblem), [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [bool](https://autolyap.github.io/dev/dev_external_reference_targets/#bool), [SolverOptions](https://autolyap.github.io/solver_backends/#autolyap.SolverOptions), [str](https://autolyap.github.io/dev/dev_external_reference_targets/#str), [Any](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Any) Search for an iteration-independent Lyapunov certificate via an SDP. Given an inclusion problem, an algorithm, and user-specified targets \((P,p,T,t,\rho,h,\alpha)\), this method formulates and solves a semidefinite feasibility problem for certificate variables \((Q,q,S,s)\). **Connection to theory** For the formal statement of the quadratic Lyapunov inequality, i.e., conditions [(C1)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c1)-[(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4), and the role of \((P,p,T,t,\rho,h,\alpha)\), see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). **User-specified targets** The tuple \((P,p,T,t)\) fixes the target lower bounds \(\mathcal{V}(P,p,k)\) and \(\mathcal{R}(T,t,k)\). The user is responsible for ensuring these target functions are nonnegative over the relevant iterates and problem class, i.e., \[\begin{split}\begin{align} \p{\forall k \in \naturals}& \quad \mathcal{V}(P,p,k) \geq 0, \\ \p{\forall k \in \naturals}& \quad \mathcal{R}(T,t,k) \geq 0. \end{align}\end{split}\] Built-in constructors for choosing \((P,p,T,t)\) are: - Linear convergence: - [`get_parameters_distance_to_solution()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.LinearConvergence.get_parameters_distance_to_solution "autolyap.IterationIndependent.LinearConvergence.get_parameters_distance_to_solution") - [`get_parameters_function_value_suboptimality()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.LinearConvergence.get_parameters_function_value_suboptimality "autolyap.IterationIndependent.LinearConvergence.get_parameters_function_value_suboptimality") - Sublinear convergence: - [`get_parameters_fixed_point_residual()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.SublinearConvergence.get_parameters_fixed_point_residual "autolyap.IterationIndependent.SublinearConvergence.get_parameters_fixed_point_residual") - [`get_parameters_duality_gap()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.SublinearConvergence.get_parameters_duality_gap "autolyap.IterationIndependent.SublinearConvergence.get_parameters_duality_gap") - [`get_parameters_function_value_suboptimality()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.SublinearConvergence.get_parameters_function_value_suboptimality "autolyap.IterationIndependent.SublinearConvergence.get_parameters_function_value_suboptimality") - [`get_parameters_optimality_measure()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.SublinearConvergence.get_parameters_optimality_measure "autolyap.IterationIndependent.SublinearConvergence.get_parameters_optimality_measure") The SDP then searches for a feasible certificate \((Q,q,S,s)\) consistent with those targets. **Parameters** - prob (`Type`[[`InclusionProblem`](https://autolyap.github.io/problem_class/#autolyap.problemclass.InclusionProblem "autolyap.problemclass.InclusionProblem")]): An [`InclusionProblem`](https://autolyap.github.io/problem_class/#autolyap.problemclass.InclusionProblem "autolyap.problemclass.InclusionProblem") instance containing interpolation conditions. - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm") instance providing dimensions and methods to compute matrices. - P ([`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")): Candidate symmetric matrix corresponding to \(P \in \sym^{n + (h+1)\NumEval + m}\). - T ([`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")): Candidate symmetric matrix corresponding to \(T \in \sym^{n + (h+\alpha+2)\NumEval + m}\). - p ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]): Candidate vector corresponding to \(p \in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc}\) for functional components (required if \(\NumFunc > 0\)). - t ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]): Candidate vector corresponding to \(t \in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}\) for functional components (required if \(\NumFunc > 0\)). - rho (`float`): A scalar contraction parameter corresponding to \(\rho\) used in forming the Lyapunov inequality (typically \(\rho \in [0,1]\)). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Nonnegative integer corresponding to \(h\) defining history. - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Nonnegative integer corresponding to \(\alpha\) defining overlap. - Q\_equals\_P ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): If True, sets Q equal to P. - S\_equals\_T ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): If True, sets S equal to T. - q\_equals\_p ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): For functional components, if True, sets q equal to p. - s\_equals\_t ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): For functional components, if True, sets s equal to t. - remove\_C2 ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): Flag to remove [(C2)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c2). - remove\_C3 ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): Flag to remove [(C3)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c3). - remove\_C4 ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): Flag to remove [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4). - solver\_options ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`SolverOptions`](https://autolyap.github.io/solver_backends/#autolyap.SolverOptions "autolyap.solver_options.SolverOptions")]): Optional backend and parameter settings. Defaults to SolverOptions(backend=”mosek\_fusion”). - verbosity ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Nonnegative output level. Defaults to 1. Set 0 to disable user-facing diagnostics, 1 for concise summaries, and 2 for per-constraint detail. **Returns** - (`Mapping`[[`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str"), [`Any`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Any "typing.Any")]): Result mapping with keys status, solve\_status, rho, and certificate. - status ([`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str")): One of “feasible”, “infeasible”, or “not\_solved”. - solve\_status ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str")]): Raw backend solve status (None when unavailable). - rho (`float`): The input contraction factor. - certificate ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[`Mapping`[[`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str"), [`Any`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Any "typing.Any")]]): None unless status == “feasible”. When status == “feasible”, certificate has: ```text { "Q": np.ndarray, # full symmetric matrix "S": np.ndarray, # full symmetric matrix "q": np.ndarray | None, # None when m_func == 0 "s": np.ndarray | None, # None when m_func == 0 "multipliers": { "operator_lambda": List[record], "function_lambda": List[record], "function_nu": List[record] } } ``` Each multiplier list entry is a record with: ```text record = { "condition": str, "component": int, "interpolation_index": int, "pairs": List[{"j": int | "star", "k": int | "star"}], "value": float } ``` Field meanings and ranges: - condition is in the active subset of [(C1)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c1), [(C2)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c2), [(C3)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c3), and [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4) (always including [(C1)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c1)). - component corresponds to \(i\) and satisfies \(i \in \llbracket 1, m \rrbracket\). - interpolation\_index corresponds to \(o\), where \(o\) is the zero-based index from \(\text{enumerate}(\text{prob.get_component_data}(i))\), so \(o \in \llbracket 0, \text{len}(\text{prob.get_component_data}(i)) - 1 \rrbracket\). - pairs is the concrete interpolation-pair list used by that multiplier. Its length is determined by [`_InterpolationIndices`](https://autolyap.github.io/dev/dev_internal_problemclass/#autolyap.problemclass.indices._InterpolationIndices "autolyap.problemclass.indices._InterpolationIndices"): 1 for “r1” and 2 for “r1 ## Linear-convergence helpers ```text static LinearConvergence.get_parameters_distance_to_solution( algo: Algorithm, h: int = 0, alpha: int = 0, i: int = 1, j: int = 1, tau: int = 0, ) → Tuple[ndarray, ndarray] | Tuple[ndarray, ndarray, ndarray, ndarray] ``` [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [Tuple](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray) Compute matrices for the distance-to-solution metric. For the matrix constructions used in this method, see [Performance estimation via SDPs](https://autolyap.github.io/theory/performance_estimation_via_sdps/). For the role of \((P,p,T,t)\), see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). **Resulting lower bounds** With this choice of \((P,p,T,t)\), \[\begin{split}\begin{aligned} \mathcal{V}(P,p,k) &= \|y_{i,j}^{k+\tau} - y^\star\|^2,\\ \mathcal{R}(T,t,k) &= 0. \end{aligned}\end{split}\] **Matrix construction** The matrix \(P\) is constructed as \[P = \left( P_{(i,j)}\, Y_\tau^{0,h} - P_{(i,\star)}\, Y_\star^{0,h} \right)^{\top} \left( P_{(i,j)}\, Y_\tau^{0,h} - P_{(i,\star)}\, Y_\star^{0,h} \right),\] where: - \(Y_\tau^{0,h}\) is the \(Y\) matrix at iteration \(\tau\) over \(\llbracket 0, h\rrbracket\), retrieved via [`_get_Ys()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ys "autolyap.algorithms.algorithm.Algorithm._get_Ys"), using k\_min = 0 and k\_max = h. - \(Y_\star^{0,h}\) is the “star” \(Y\) matrix over \(\llbracket 0, h\rrbracket\), retrieved via [`_get_Ys()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ys "autolyap.algorithms.algorithm.Algorithm._get_Ys"), using k\_min = 0 and k\_max = h. - \(P_{(i,j)}\) and \(P_{(i,\star)}\) are the projection matrices for component \(i\), retrieved via [`_get_Ps()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ps "autolyap.algorithms.algorithm.Algorithm._get_Ps"). The remaining matrices and vectors are set to zero: - \(T = 0\). - If \(\NumFunc > 0\), then \(p = 0\) and \(t = 0\). **Parameters** - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An instance of [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm"). It must provide algo.m, algo.m\_bar\_is, [`_get_Ys()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ys "autolyap.algorithms.algorithm.Algorithm._get_Ys"), and [`_get_Ps()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ps "autolyap.algorithms.algorithm.Algorithm._get_Ps"). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(h\) defining the time horizon \(\llbracket 0, h\rrbracket\) for \(Y\) matrices. - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(\alpha\) for extending the horizon for \(T\) (and \(t\)). - i ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Component index (1-indexed) corresponding to \(i\). Default is 1; must satisfy \(i \in \llbracket 1, m\rrbracket\), where m = algo.m. - j ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Evaluation index for component i corresponding to \(j\). Default is 1; must satisfy \(j \in \llbracket 1, \NumEval_i\rrbracket\), where \(\NumEval_i\) is given by algo.m\_bar\_is[i-1]. - tau ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Iteration index corresponding to \(\tau\). Default is 0; must satisfy \(\tau \in \llbracket 0, h\rrbracket\). **Returns** - ([`Union`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Union "typing.Union")[[`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")], [`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]]): If algo.m\_func == 0, returns (P, T) with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m}. \end{aligned}\end{split}\] Otherwise, returns (P, p, T, t), where \(P\) is computed as above and \(T\), \(p\), and \(t\) are zero arrays with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m},\\ p &\in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc},\\ t &\in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}. \end{aligned}\end{split}\] **Raises** - ValueError: If any input is out of its valid range or if required matrices are missing. ```text static LinearConvergence.get_parameters_function_value_suboptimality( algo: Algorithm, h: int = 0, alpha: int = 0, j: int = 1, tau: int = 0, ) → Tuple[ndarray, ndarray, ndarray, ndarray] ``` [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [Tuple](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray) Compute matrices and vectors for function-value suboptimality. For the matrix constructions used in this method, see [Performance estimation via SDPs](https://autolyap.github.io/theory/performance_estimation_via_sdps/). For the role of \((P,p,T,t)\), see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). **Resulting lower bounds** With this choice of \((P,p,T,t)\), \[\begin{split}\begin{aligned} \mathcal{V}(P,p,k) &= f_1(y_{1,j}^{k+\tau}) - f_1(y^\star),\\ \mathcal{R}(T,t,k) &= 0. \end{aligned}\end{split}\] **Matrix construction** This method applies only when \(m = \NumFunc = 1\). The vector \(p\) is constructed as \[p = \left( F_{(1,j,\tau)}^{0,h} - F_{(1,\star,\star)}^{0,h} \right)^{\top},\] where: - \(F_{(1,j,\tau)}^{0,h}\) is retrieved via [`_get_Fs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Fs "autolyap.algorithms.algorithm.Algorithm._get_Fs"), using k\_min = 0 and k\_max = h. - \(F_{(1,\star,\star)}^{0,h}\) is retrieved via [`_get_Fs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Fs "autolyap.algorithms.algorithm.Algorithm._get_Fs"), using k\_min = 0 and k\_max = h. The remaining matrices and vectors are set to zero: - \(P = 0\). - \(T = 0\). - \(t = 0\). **Parameters** - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An instance of [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm"). It must satisfy algo.m == 1, algo.m\_func == 1, and provide [`_get_Fs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Fs "autolyap.algorithms.algorithm.Algorithm._get_Fs"). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(h\) defining the horizon \(\llbracket 0, h\rrbracket\) for \(F\) matrices. - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(\alpha\) for extending the horizon for \(T\) and \(t\). - j ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Evaluation index for component 1 corresponding to \(j\). Default is 1; must satisfy \(j \in \llbracket 1, \NumEval_1\rrbracket\), where \(\NumEval_1\) is given by algo.m\_bar\_is[0]. - tau ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Iteration index corresponding to \(\tau\). Default is 0; must satisfy \(\tau \in \llbracket 0, h\rrbracket\). **Returns** - ([`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]): A tuple \((P, p, T, t)\), where \(p\) is computed as above (a one-dimensional NumPy array), and \(P\), \(T\), and \(t\) are zero arrays, with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m},\\ p &\in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc},\\ t &\in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}. \end{aligned}\end{split}\] **Raises** - ValueError: If algo.m != 1 or algo.m\_func != 1, if any input parameter is out of range, or if the required \(F\) matrices are not found. ```text static LinearConvergence.bisection_search_rho( prob: InclusionProblem, algo: Algorithm, P: ndarray, T: ndarray, p: ndarray | None = None, t: ndarray | None = None, h: int = 0, alpha: int = 0, Q_equals_P: bool = False, S_equals_T: bool = False, q_equals_p: bool = False, s_equals_t: bool = False, remove_C2: bool = False, remove_C3: bool = False, remove_C4: bool = True, lower_bound: float = 0.0, upper_bound: float = 1.0, tol: float = 1e-12, solver_options: SolverOptions | None = None, verbosity: int = 1, ) → Mapping[str, Any] ``` [InclusionProblem](https://autolyap.github.io/problem_class/#autolyap.problemclass.InclusionProblem), [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [bool](https://autolyap.github.io/dev/dev_external_reference_targets/#bool), [SolverOptions](https://autolyap.github.io/solver_backends/#autolyap.SolverOptions), [str](https://autolyap.github.io/dev/dev_external_reference_targets/#str), [Any](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Any) Perform a bisection search to find the minimum contraction parameter \(\rho\). This method performs a bisection search over \(\rho\) in the interval \([{\text{lower_bound}}, {\text{upper_bound}}]\) to find the minimal value for which the iteration-independent Lyapunov inequality holds. At each step it re-solves the same model with an updated \(\rho\) until the interval size is below \({\text{tol}}\). Each feasibility check is performed by [`search_lyapunov()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.search_lyapunov "autolyap.IterationIndependent.search_lyapunov"); see its documentation for the enforced SDP feasibility checks. **Parameters** - prob (`Type`[[`InclusionProblem`](https://autolyap.github.io/problem_class/#autolyap.problemclass.InclusionProblem "autolyap.problemclass.InclusionProblem")]): An [`InclusionProblem`](https://autolyap.github.io/problem_class/#autolyap.problemclass.InclusionProblem "autolyap.problemclass.InclusionProblem") instance containing interpolation conditions. - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm") instance providing dimensions and methods. - P ([`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")): A symmetric matrix corresponding to \(P \in \sym^{n + (h+1)\NumEval + m}\). - T ([`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")): A symmetric matrix corresponding to \(T \in \sym^{n + (h+\alpha+2)\NumEval + m}\). - p ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]): A vector corresponding to \(p \in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc}\) for functional components (if applicable). - t ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]): A vector corresponding to \(t \in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}\) for functional components (if applicable). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Nonnegative integer corresponding to \(h\) defining the history for the matrices. - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Nonnegative integer corresponding to \(\alpha\) for extending the horizon. - Q\_equals\_P ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): If True, set Q equal to P. - S\_equals\_T ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): If True, set S equal to T. - q\_equals\_p ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): For functional components, if True, set q equal to p. - s\_equals\_t ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): For functional components, if True, set s equal to t. - remove\_C2 ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): Flag to remove [(C2)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c2). - remove\_C3 ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): Flag to remove [(C3)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c3). - remove\_C4 ([`bool`](https://autolyap.github.io/dev/dev_external_reference_targets/#bool "bool")): Flag to remove [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4). - lower\_bound (`float`): Lower bound for \(\rho\). - upper\_bound (`float`): Upper bound for \(\rho\). - tol (`float`): Tolerance for the bisection search stopping criterion. - solver\_options ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`SolverOptions`](https://autolyap.github.io/solver_backends/#autolyap.SolverOptions "autolyap.solver_options.SolverOptions")]): Optional backend and parameter settings. Defaults to SolverOptions(backend=”mosek\_fusion”). - verbosity ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Nonnegative output level. Defaults to 1. Set 0 to disable user-facing diagnostics, 1 for concise summaries, and 2 for per-constraint detail. **Returns** - (`Mapping`[[`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str"), [`Any`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Any "typing.Any")]): Result mapping with keys status, solve\_status, rho, and certificate. - status ([`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str")): One of “feasible”, “infeasible”, or “not\_solved”. - solve\_status ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[[`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str")]): Raw backend solve status for the terminal check (None when unavailable). - rho ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[`float`]): Best feasible bisection value when status == “feasible”; otherwise None. - certificate ([`Optional`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Optional "typing.Optional")[`Mapping`[[`str`](https://autolyap.github.io/dev/dev_external_reference_targets/#str "str"), [`Any`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Any "typing.Any")]]): Feasibility certificate when status == “feasible”; otherwise None. The certificate schema is exactly the same as in [`search_lyapunov()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.search_lyapunov "autolyap.IterationIndependent.search_lyapunov"). **Raises** - ValueError: If any input is out of range or the bounds are inconsistent. - mosek.fusion.OptimizeError: If the MOSEK backend is selected and raises a license-related error during optimization. ## Sublinear-convergence helpers ```text static SublinearConvergence.get_parameters_fixed_point_residual( algo: Algorithm, h: int = 0, alpha: int = 0, tau: int = 0, ) → Tuple[ndarray, ndarray] | Tuple[ndarray, ndarray, ndarray, ndarray] ``` [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [Tuple](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray) Compute matrices for the fixed-point residual. For the matrix constructions used in this method, see [Performance estimation via SDPs](https://autolyap.github.io/theory/performance_estimation_via_sdps/). For the role of \((P,p,T,t)\), see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). **Resulting lower bounds** For iteration index \(\tau\) (with \(\tau \in \llbracket 0, h+\alpha+1\rrbracket\)), this choice of \((P,p,T,t)\) gives: \[\begin{split}\begin{aligned} \mathcal{V}(P,p,k) &= 0,\\ \mathcal{R}(T,t,k) &= \|\bx^{k+\tau+1} - \bx^{k+\tau}\|^2. \end{aligned}\end{split}\] **Matrix construction** The matrix \(T\) is constructed as \[T = \left( X_{\tau+1}^{0, h+\alpha+1} - X_{\tau}^{0, h+\alpha+1} \right)^{\top} \left( X_{\tau+1}^{0, h+\alpha+1} - X_{\tau}^{0, h+\alpha+1} \right),\] where: - \(X_{\tau}^{0,h+\alpha+1}\) is retrieved via [`_get_Xs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Xs "autolyap.algorithms.algorithm.Algorithm._get_Xs"), using k\_min = 0 and k\_max = h+alpha+1. - \(X_{\tau+1}^{0,h+\alpha+1}\) is retrieved via [`_get_Xs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Xs "autolyap.algorithms.algorithm.Algorithm._get_Xs"), using k\_min = 0 and k\_max = h+alpha+1. The remaining matrices and vectors are set to zero: - \(P = 0\). - If \(\NumFunc > 0\), then \(p = 0\) and \(t = 0\). **Parameters** - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An instance of [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm"). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(h\) defining the time horizon \(\llbracket 0, h\rrbracket\) for \(P\). - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(\alpha\) for extending the horizon for \(T\) (and \(t\)). - tau ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Iteration index corresponding to \(\tau\) for computing the fixed-point residual. Must satisfy \(\tau \in \llbracket 0, h+\alpha+1\rrbracket\). **Returns** - ([`Union`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Union "typing.Union")[[`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")], [`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]]): If algo.m\_func == 0, returns (P, T) with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m}. \end{aligned}\end{split}\] Otherwise, returns (P, p, T, t) with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m},\\ p &\in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc},\\ t &\in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}. \end{aligned}\end{split}\] **Raises** - ValueError: If any input parameter is out of its valid range or if the required \(X\) matrices are missing. ```text static SublinearConvergence.get_parameters_duality_gap( algo: Algorithm, h: int = 0, alpha: int = 0, tau: int = 0, ) → Tuple[ndarray, ndarray, ndarray, ndarray] ``` [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [Tuple](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray) Compute matrices for the duality gap. For the matrix constructions used in this method, see [Performance estimation via SDPs](https://autolyap.github.io/theory/performance_estimation_via_sdps/). For the role of \((P,p,T,t)\), see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). **Resulting lower bounds** For iteration index \(\tau\) (with \(\tau \in \llbracket 0, h+\alpha+1\rrbracket\)), Warning TODO. **Matrix construction** The matrix \(T\) and vector \(t\) are constructed as \[\begin{split}T = -\frac{1}{2} \sum_{i=1}^{m} \begin{bmatrix} P_{(i,\star)}\, U_\star^{0,h+\alpha+1} \\ P_{(i,1)}\, Y_\tau^{0,h+\alpha+1} \end{bmatrix}^{\top} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} P_{(i,\star)}\, U_\star^{0,h+\alpha+1} \\ P_{(i,1)}\, Y_\tau^{0,h+\alpha+1} \end{bmatrix},\end{split}\] and \[t = \sum_{i=1}^{m} \left( F_{(i,1,\tau)}^{0,h+\alpha+1} - F_{(i,\star,\star)}^{0,h+\alpha+1} \right)^{\top}.\] where: - \(U_\star^{0,h+\alpha+1}\) is retrieved via [`_get_Us()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Us "autolyap.algorithms.algorithm.Algorithm._get_Us"), using k\_min = 0 and k\_max = h+alpha+1. - \(Y_\tau^{0,h+\alpha+1}\) is retrieved via [`_get_Ys()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ys "autolyap.algorithms.algorithm.Algorithm._get_Ys"), using k\_min = 0 and k\_max = h+alpha+1. - \(F_{(i,1,\tau)}^{0,h+\alpha+1}\) and \(F_{(i,\star,\star)}^{0,h+\alpha+1}\) are retrieved via [`_get_Fs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Fs "autolyap.algorithms.algorithm.Algorithm._get_Fs"), using k\_min = 0 and k\_max = h+alpha+1. - \(P_{(i,\star)}\) and \(P_{(i,1)}\) are retrieved via [`_get_Ps()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ps "autolyap.algorithms.algorithm.Algorithm._get_Ps"). The remaining matrices and vectors are set to zero: - \(P = 0\). - \(p = 0\). **Requirements** Requires \(m = \NumFunc\) (i.e., all components are functional). **Parameters** - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An instance of [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm") (with algo.m == algo.m\_func). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(h\) defining the time horizon \(\llbracket 0, h\rrbracket\) for \(P\). - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(\alpha\) for extending the horizon for \(T\) and \(t\). - tau ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Iteration index corresponding to \(\tau\) for computing the duality gap. Must satisfy \(\tau \in \llbracket 0, h+\alpha+1\rrbracket\). **Returns** - ([`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]): A tuple \((P, p, T, t)\), where \(t\) is a one-dimensional NumPy array, with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m},\\ p &\in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc},\\ t &\in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}. \end{aligned}\end{split}\] **Raises** - ValueError: If any input parameter is out of its valid range, if required matrices are missing, or if \(m \ne \NumFunc\). ```text static SublinearConvergence.get_parameters_function_value_suboptimality( algo: Algorithm, h: int = 0, alpha: int = 0, j: int = 1, tau: int = 0, ) → Tuple[ndarray, ndarray, ndarray, ndarray] ``` [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [Tuple](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray) Compute matrices and vectors for function-value suboptimality. For the matrix constructions used in this method, see [Performance estimation via SDPs](https://autolyap.github.io/theory/performance_estimation_via_sdps/). For the role of \((P,p,T,t)\), see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). **Resulting lower bounds** With this choice of \((P,p,T,t)\), \[\begin{split}\begin{aligned} \mathcal{V}(P,p,k) &= 0,\\ \mathcal{R}(T,t,k) &= f_1(y_{1,j}^{k+\tau}) - f_1(y^\star). \end{aligned}\end{split}\] **Matrix construction** This method applies only when \(m = \NumFunc = 1\). The vector \(t\) is constructed as \[t = \left( F_{(1,j,\tau)}^{0,h+\alpha+1} - F_{(1,\star,\star)}^{0,h+\alpha+1} \right)^{\top},\] where: - \(F_{(1,j,\tau)}^{0,h+\alpha+1}\) is retrieved via [`_get_Fs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Fs "autolyap.algorithms.algorithm.Algorithm._get_Fs"), using k\_min = 0 and k\_max = h+alpha+1. - \(F_{(1,\star,\star)}^{0,h+\alpha+1}\) is retrieved via [`_get_Fs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Fs "autolyap.algorithms.algorithm.Algorithm._get_Fs"), using k\_min = 0 and k\_max = h+alpha+1. The remaining matrices and vectors are set to zero: - \(P = 0\). - \(T = 0\). - \(p = 0\). **Parameters** - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An instance of [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm"). It must satisfy algo.m == 1, algo.m\_func == 1, and provide [`_get_Fs()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Fs "autolyap.algorithms.algorithm.Algorithm._get_Fs"). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(h\) defining the horizon \(\llbracket 0, h + \alpha + 1\rrbracket\) for \(F\) matrices. - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(\alpha\) for extending the horizon for \(T\) and \(t\). - j ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Evaluation index for component 1 corresponding to \(j\). Default is 1; must satisfy \(j \in \llbracket 1, \NumEval_1\rrbracket\), where \(\NumEval_1\) is given by algo.m\_bar\_is[0]. - tau ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Iteration index corresponding to \(\tau\). Default is 0; must satisfy \(\tau \in \llbracket 0, h + \alpha + 1\rrbracket\). **Returns** - ([`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]): A tuple \((P, p, T, t)\), where \(t\) is computed as above (a one-dimensional NumPy array), and \(P\), \(T\), and \(p\) are zero arrays, with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m},\\ p &\in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc},\\ t &\in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}. \end{aligned}\end{split}\] **Raises** - ValueError: If algo.m != 1 or algo.m\_func != 1, if any input parameter is out of range, or if the required \(F\) matrices are not found. ```text static SublinearConvergence.get_parameters_optimality_measure( algo: Algorithm, h: int = 0, alpha: int = 0, tau: int = 0, ) → Tuple[ndarray, ndarray] | Tuple[ndarray, ndarray, ndarray, ndarray] ``` [Algorithm](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm), [int](https://autolyap.github.io/dev/dev_external_reference_targets/#int), [Tuple](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple), [ndarray](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray) Compute matrices for the optimality measure. For the matrix constructions used in this method, see [Performance estimation via SDPs](https://autolyap.github.io/theory/performance_estimation_via_sdps/). For the role of \((P,p,T,t)\), see [Iteration-independent analyses](https://autolyap.github.io/theory/iteration_independent_analyses/). **Resulting lower bounds** For iteration index \(\tau\) (with \(\tau \in \llbracket 0, h+\alpha+1\rrbracket\)), this choice of \((P,p,T,t)\) gives: \[\mathcal{V}(P,p,k) = 0,\] and \[\begin{split}\mathcal{R}(T,t,k) = \begin{cases} \|u_{1,1}^{k+\tau}\|^2 & \text{if } m = 1, \\ \left\|\sum_{i=1}^{m} u_{i,1}^{k+\tau}\right\|^2 + \sum_{i=2}^{m} \|y_{1,1}^{k+\tau} - y_{i,1}^{k+\tau}\|^2 & \text{if } m > 1. \end{cases}\end{split}\] **Matrix construction** The matrix \(T\) is constructed as \[\begin{split}T = \begin{cases} \left( P_{(1,1)}\, U_\tau^{0,h+\alpha+1} \right)^{\top} \left( P_{(1,1)}\, U_\tau^{0,h+\alpha+1} \right) & \text{if } m = 1, \\[1em] \left( \left( \sum_{i=1}^{m} P_{(i,1)}\, U_\tau^{0,h+\alpha+1} \right)^{\top} \left( \sum_{i=1}^{m} P_{(i,1)}\, U_\tau^{0,h+\alpha+1} \right) + \sum_{i=2}^{m} \left( \left( P_{(1,1)} - P_{(i,1)} \right) Y_\tau^{0,h+\alpha+1} \right)^{\top} \left( \left( P_{(1,1)} - P_{(i,1)} \right) Y_\tau^{0,h+\alpha+1} \right) \right) & \text{if } m > 1, \end{cases}\end{split}\] where: - \(U_\tau^{0,h+\alpha+1}\) is retrieved via [`_get_Us()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Us "autolyap.algorithms.algorithm.Algorithm._get_Us"), using k\_min = 0 and k\_max = h+alpha+1. - \(Y_\tau^{0,h+\alpha+1}\) is retrieved via [`_get_Ys()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ys "autolyap.algorithms.algorithm.Algorithm._get_Ys"), using k\_min = 0 and k\_max = h+alpha+1. - \(P_{(i,1)}\) are projection matrices retrieved via [`_get_Ps()`](https://autolyap.github.io/dev/dev_internal_algorithms/#autolyap.algorithms.algorithm.Algorithm._get_Ps "autolyap.algorithms.algorithm.Algorithm._get_Ps"). The remaining matrices and vectors are set to zero: - \(P = 0\). - If \(\NumFunc > 0\), then \(p = 0\) and \(t = 0\). **Parameters** - algo (`Type`[[`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm")]): An instance of [`Algorithm`](https://autolyap.github.io/base_algorithms/#autolyap.algorithms.Algorithm "autolyap.algorithms.Algorithm"). - h ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(h\) defining the time horizon \(\llbracket 0, h\rrbracket\) for \(P\). - alpha ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): A nonnegative integer corresponding to \(\alpha\) for extending the horizon for \(T\) (and \(t\)). - tau ([`int`](https://autolyap.github.io/dev/dev_external_reference_targets/#int "int")): Iteration index corresponding to \(\tau\) for computing the optimality measure. Must satisfy \(\tau \in \llbracket 0, h+\alpha+1\rrbracket\). **Returns** - ([`Union`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Union "typing.Union")[[`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")], [`Tuple`](https://autolyap.github.io/dev/dev_external_reference_targets/#typing.Tuple "typing.Tuple")[[`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray"), [`numpy.ndarray`](https://autolyap.github.io/dev/dev_external_reference_targets/#numpy.ndarray "numpy.ndarray")]]): If algo.m\_func == 0, returns (P, T) with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m}. \end{aligned}\end{split}\] Otherwise, returns (P, p, T, t) with \[\begin{split}\begin{aligned} P &\in \sym^{n + (h+1)\NumEval + m},\\ T &\in \sym^{n + (h+\alpha+2)\NumEval + m},\\ p &\in \mathbb{R}^{(h+1)\NumEvalFunc + \NumFunc},\\ t &\in \mathbb{R}^{(h+\alpha+2)\NumEvalFunc + \NumFunc}. \end{aligned}\end{split}\] **Raises** - ValueError: If any input parameter is out of its valid range or if required matrices are missing. TeX macro definitions used in this page: ```latex \newcommand{\NumEval}{\bar{m}} \newcommand{\NumEvalFunc}{\bar{m}_{\textup{func}}} \newcommand{\NumFunc}{m_{\textup{func}}} \newcommand{\bx}{\mathbf{x}} \newcommand{\llbracket}{\lbrack\!\lbrack} \newcommand{\naturals}{\mathbb{N}_{0}} \newcommand{\p}[1]{\mathord{( #1 )}} \newcommand{\rrbracket}{\rbrack\!\rbrack} \newcommand{\sym}{\mathbb{S}} ```