Canonical HTML: https://autolyap.github.io/theory/iteration_independent_analyses/ Documentation version: 0.2.1 # 5.2. Iteration-independent analyses For iteration-independent analyses, we consider algorithms that continue to iterate indefinitely and have iteration-independent parameters. I.e., in [(3.1)](https://autolyap.github.io/theory/algorithm_representation/#equation-eq-linear-system-with-nonlinearity), we assume that \(K=\infty\) and that there exist fixed matrices \[\begin{aligned} \p{A,B,C,D}\in\reals^{n\times n}\times\reals^{n\times \NumEval}\times\reals^{\NumEval\times n}\times\reals^{\NumEval \times \NumEval}\end{aligned}\] such that (5.24)\[\begin{aligned} \p{\forall k \in \naturals }\quad \p{A_{k},B_{k},C_{k},D_{k}} = \p{A,B,C,D}.\end{aligned}\] **Definition 5.2.1 (Iteration-independent quadratic Lyapunov inequality).** Suppose that [Assumption 3.1 (Well-posedness)](https://autolyap.github.io/theory/algorithm_representation/#ass-well-posedness) and [(5.24)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-constant-abcd) hold. Let \[\bx_{0}\in\calH^{n}, \qquad \p{ ( \bx^{k}, \bu^{k}, \by^{k}, \bFcn^{k} ) }_{k\in\naturals},\] be an initial point and a sequence of iterates satisfying [(3.1)](https://autolyap.github.io/theory/algorithm_representation/#equation-eq-linear-system-with-nonlinearity), and let \[\p{y^{\star}, \hat{\bu}^{\star}, \bFcn^{\star}}\] be a point satisfying [(5.1)](https://autolyap.github.io/theory/lyapunov_analyses/#equation-eq-solution). Also let \[\rho \in [0,1], \qquad h\in\naturals, \qquad \alpha\in\naturals.\] Define (5.25)\[\begin{split}\begin{aligned} \p{\forall k \in \naturals} \quad \mathcal{V}(W,w,k) {}={}&{} \quadform{W}{\p{\bx^{k},\bu^{k},\ldots,\bu^{k+h},\hat{\bu}^{\star},y^{\star}}} \\ & + w^{\top}\p{\bFcn^{k},\ldots,\bFcn^{k+h},\bFcn^{\star}}, \end{aligned}\end{split}\] for each \(\p{W,w}\in\set{\p{Q,q},\p{P,p}}\), where \[Q,P \in \sym^{n+\p{h+1}\NumEval+m}, \qquad q,p\in\mathbb{R}^{\p{h+1}\NumEvalFunc + \NumFunc},\] and define (5.26)\[\begin{split}\begin{aligned} \p{\forall k \in \naturals} \quad \mathcal{R}(W,w,k) {}={}&{} \quadform{W}{\p{\bx^{k},\bu^{k},\ldots,\bu^{k+h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} \\ &{} + w^{\top}\p{\bFcn^{k},\ldots,\bFcn^{k+h + \alpha + 1},\bFcn^{\star}}, \end{aligned}\end{split}\] for each \(\p{W,w}\in\set{\p{S,s},\p{T,t}}\), where \[S,T\in\sym^{n+\p{h+\alpha+2}\NumEval+m}, \qquad s, t \in\mathbb{R}^{\p{h+\alpha+2}\NumEvalFunc + \NumFunc}.\] We say that \(\p{Q,q,S,s}\) satisfies a \(\p{P,p,T,t,\rho, h, \alpha}\)-*quadratic Lyapunov inequality* for algorithm [(3.1)](https://autolyap.github.io/theory/algorithm_representation/#equation-eq-linear-system-with-nonlinearity) over the problem class defined by \((\mathcal{F}_i)_{i\in\IndexFunc}\) and \((\mathcal{G}_i)_{i\in\IndexOp}\) if \begin{align} \p{\forall k \in \naturals} &\quad \mathcal{V}\p{Q,q,k+\alpha+1} \leq \rho \mathcal{V}\p{Q,q,k} - \mathcal{R}\p{S,s,k}, \tag{C1} \\ \p{\forall k \in \naturals} &\quad \mathcal{V}\p{Q,q,k} \geq \mathcal{V}\p{P,p,k} \geq 0, \tag{C2} \\ \p{\forall k \in \naturals} &\quad \mathcal{R}\p{S,s,k} \geq \mathcal{R}\p{T,t,k}\geq 0, \tag{C3} \\ \p{\forall k \in \naturals} &\quad \mathcal{R}\p{S,s,k+1} \leq \mathcal{R}\p{S,s,k}, \tag{C4} \end{align} hold for each \[\bx_{0}, \qquad \p{ ( \bx^{k}, \bu^{k}, \by^{k}, \bFcn^{k} ) }_{k\in\naturals}, \qquad \p{y^{\star}, \hat{\bu}^{\star},\bFcn^{\star}},\] and for each \[\p{f_{i}}_{i\in\IndexFunc} \in \prod_{i\in\IndexFunc} \mathcal{F}_{i}, \qquad \p{G_i}_{i\in\IndexOp} \in \prod_{i\in\IndexOp} \mathcal{G}_i,\] where [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4) is an optional requirement that may be removed. Definition 5.2.1 is a trajectory-level condition: [(C1)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c1)-[(C3)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c3) (and optionally [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4)) must hold for all iterates and all admissible functions/operators in the class. This statement is not directly computational. In practice, the user specifies \(\p{P,p,T,t,\rho,h,\alpha}\) and searches for \(\p{Q,q,S,s}\). The SDP sufficient condition that makes this search computable is given in [Theorem 5.2.2 (Iteration-independent Lyapunov inequality via SDP)](https://autolyap.github.io/theory/iteration_independent_analyses/#thm-iteration-independent-lyapunov). When such \(\p{Q,q,S,s}\) exists, the choice of \(\p{P,p,T,t,\rho,h,\alpha}\) determines the convergence guarantee. In particular, we obtain: - Linear setting: if \(\rho \in [0,1[\), then \[\begin{aligned} 0 \leq \mathcal{V}\p{P,p,k} \leq \mathcal{V}\p{Q,q,k} \leq \rho^{\lfloor k/\p{\alpha+1}\rfloor }\max_{i\in\llbracket0,\alpha\rrbracket}\mathcal{V}\p{Q,q,i} \xrightarrow[k\rightarrow \infty]{} 0. \end{aligned}\] Thus, \((\mathcal{V}\p{P,p,k})_{k\in\naturals}\) converges to zero \[\sqrt[\alpha + 1]{\rho}\textup{-linearly}.\] - Sublinear setting: if \(\rho = 1\), then \[\begin{aligned} \p{\forall k \in \naturals}\quad \sum_{i=0}^{k} \mathcal{R}\p{T,t,i} \leq \sum_{i=0}^{k} \mathcal{R}\p{S,s,i} \leq \sum_{j=0}^{\alpha} \mathcal{V}\p{Q,q,j}, \end{aligned}\] by a telescoping summation argument. In particular, \[(\mathcal{R}\p{T,t,k})_{k\in\naturals}\] is summable, converges to zero, and, e.g., \[\min_{i\in \llbracket 0,k \rrbracket } \mathcal{R}\p{T,t,i} \in \mathcal{O}\p{1/k} \quad \textup{ as } \quad k\to\infty.\] If the optional requirement [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4) holds, we obtain the stronger last-iterate convergence result \[\mathcal{R}\p{T,t,k} \in o\p{1/k} \quad \textup{ as } \quad k\to\infty.\] The user-chosen \(\p{P,p,T,t}\) fixes the particular Lyapunov and residual expressions \(\mathcal{V}\p{P,p,\cdot}\) and \(\mathcal{R}\p{T,t,\cdot}\) to be analyzed. For built-in choices: - Linear convergence: [Linear-convergence helpers](https://autolyap.github.io/iteration_independent_analysis/#iter-independent-linear-helpers). - Sublinear convergence: [Sublinear-convergence helpers](https://autolyap.github.io/iteration_independent_analysis/#iter-independent-sublinear-helpers). For the corresponding API entry points, use [`search_lyapunov()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.search_lyapunov "autolyap.IterationIndependent.search_lyapunov") for fixed \(\rho\), and [`bisection_search_rho()`](https://autolyap.github.io/iteration_independent_analysis/#autolyap.IterationIndependent.LinearConvergence.bisection_search_rho "autolyap.IterationIndependent.LinearConvergence.bisection_search_rho") to estimate the smallest feasible contraction factor in the linear-convergence setting. Role of \(h\) and \(\alpha\): - \(h\) is a history parameter for \(\mathcal{V}\) (see [(5.25)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-v)). \[\text{the } \mathcal{V}\text{-window length is } h+1.\] - \(\alpha\) is an overlap parameter that induces the shift \(k \mapsto k+\alpha+1\) in [(C1)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c1). - \(h\) and \(\alpha\) determine the \(\mathcal{R}\) window (see [(5.26)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-r)). \[\text{the } \mathcal{R}\text{-window length is } h+\alpha+2.\] We now state the finite-dimensional SDP condition that certifies the existence of such \(\p{Q,q,S,s}\). **Theorem 5.2.2 (Iteration-independent Lyapunov inequality via SDP).** Suppose that [Assumption 3.1 (Well-posedness)](https://autolyap.github.io/theory/algorithm_representation/#ass-well-posedness), [Assumption 4.1 (Interpolation conditions)](https://autolyap.github.io/theory/interpolation_conditions/#ass-interpolation) and [(5.24)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-constant-abcd) hold. Let \[\rho \in [0,1], \qquad h\in\naturals, \qquad \alpha\in\naturals,\] and let \[P \in \sym^{n+\p{h+1}\NumEval+m}, \qquad p\in\mathbb{R}^{\p{h+1}\NumEvalFunc + \NumFunc},\] such that \[\begin{aligned} \p{\forall k \in \naturals} \qquad \mathcal{V}\p{P,p,k} \geq 0, \end{aligned}\] where \(\mathcal{V}\) is defined in [(5.25)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-v), and let \[T\in\sym^{n+\p{h+\alpha+2}\NumEval+m}, \qquad t\in\mathbb{R}^{\p{h+\alpha+2}\NumEvalFunc + \NumFunc},\] such that \[\begin{aligned} \p{\forall k \in \naturals} \qquad \mathcal{R}\p{T,t,k} \geq 0, \end{aligned}\] where \(\mathcal{R}\) is defined in [(5.26)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-r). Then a sufficient condition for there to exist \(\p{Q,q,S,s}\) that satisfies a \(\p{P,p,T,t,\rho, h, \alpha}\)-quadratic Lyapunov inequality for algorithm [(3.1)](https://autolyap.github.io/theory/algorithm_representation/#equation-eq-linear-system-with-nonlinearity) over the problem class defined by \(\p{\mathcal{F}_i}_{i\in\IndexFunc}\) and \(\p{\mathcal{G}_i}_{i\in\IndexOp}\) (recall that [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4) is optional and may be omitted) is that the following system of constraints, which reuses the lifted matrices/vectors from [5.1. Performance estimation via SDPs](https://autolyap.github.io/theory/performance_estimation_via_sdps/), including the lifted state matrices [(5.3)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#equation-eq-x-mats), and the interpolation terms [(5.12)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#equation-eq-w-func-ineq), [(5.13)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#equation-eq-w-func-eq), [(5.14)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#equation-eq-w-op), [(5.15)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#equation-eq-f-func-ineq), and [(5.16)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#equation-eq-f-func-eq), \begin{align} & \textbf{for each}\ \textup{cond} \in\set{\href{#eq-c1}{\textup{C1}},\href{#eq-c2}{\textup{C2}},\href{#eq-c3}{\textup{C3}},\href{#eq-c4}{\textup{C4}}} \notag \\ &\qquad \left( \begin{array}{@{}c@{}} \forall i \in\IndexFunc \\ \forall o \in \mathcal{O}^{\textup{func-ineq}}_{i} \\ \forall \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} \end{array} \right) \qquad \lambda_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-ineq},\,\textup{cond}} \geq 0, \tag{5.28a}\\ &\qquad \left( \begin{array}{@{}c@{}} \forall i \in\IndexFunc \\ \forall o \in \mathcal{O}^{\textup{func-eq}}_{i} \\ \forall \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} \end{array} \right) \qquad \nu_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-eq},\,\textup{cond}} \in \reals, \tag{5.28b}\\ &\qquad \left( \begin{array}{@{}c@{}} \forall i \in\IndexOp \\ \forall o \in \mathcal{O}^{\textup{op}}_{i} \\ \forall \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} \end{array} \right) \qquad \lambda_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{op},\,\textup{cond}} \geq 0, \tag{5.28c}\\ &\qquad - W^{\textup{cond}} + \sum_{\substack{i\in\IndexFunc \\ o \in \mathcal{O}^{\textup{func-ineq}}_{i} \\ \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} }} \lambda_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-ineq},\,\textup{cond}} W_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-ineq}} \notag \\ &\qquad \qquad + \sum_{\substack{i\in\IndexFunc \\ o \in \mathcal{O}^{\textup{func-eq}}_{i} \\ \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} }} \nu_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-eq},\,\textup{cond}} W_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-eq}} \notag \\ &\qquad \qquad + \sum_{\substack{i\in\IndexOp \\ o \in \mathcal{O}^{\textup{op}}_{i} \\ \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} }} \lambda_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{op},\,\textup{cond}} W_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{op}} \succeq 0, \tag{5.28d}\\ &\qquad - w^{\textup{cond}} + \sum_{\substack{i\in\IndexFunc \\ o \in \mathcal{O}^{\textup{func-ineq}}_{i} \\ \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} }} \lambda_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-ineq},\,\textup{cond}} F_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-ineq}} \notag \\ &\qquad \qquad + \sum_{\substack{i\in\IndexFunc \\ o \in \mathcal{O}^{\textup{func-eq}}_{i} \\ \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} }} \nu_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-eq},\,\textup{cond}} F_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-eq}} = 0, \tag{5.28e}\\ & \mathbf{end} \notag \\ & Q \in \sym^{n+\p{h+1}\NumEval+m}, \tag{5.28f}\\ & q \in \mathbb{R}^{\p{h+1}\NumEvalFunc + \NumFunc}, \tag{5.28g}\\ & S \in \sym^{n+\p{h+\alpha+2}\NumEval+m}, \tag{5.28h}\\ & s \in \reals^{\p{h+\alpha+2}\NumEvalFunc + \NumFunc}. \tag{5.28i} \end{align} is feasible for the scalars \[\begin{split}\begin{aligned} &\lambda_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-ineq},\,\textup{cond}}, \\ &\nu_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{func-eq},\,\textup{cond}}, \\ &\lambda_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{\textup{op},\,\textup{cond}}, \end{aligned}\end{split}\] the matrices \(Q\) and \(S\), and the vectors \(q\) and \(s\), where (5.29)\[\begin{aligned} W^{\href{#eq-c1}{\textup{C1}}} &= \p{\Theta_{1}^{\href{#eq-c1}{\textup{C1}}}}^{\top}Q\Theta_{1}^{\href{#eq-c1}{\textup{C1}}} - \rho \p{\Theta_{0}^{\href{#eq-c1}{\textup{C1}}}}^{\top}Q\Theta_{0}^{\href{#eq-c1}{\textup{C1}}} + S, \end{aligned}\] (5.30)\[\begin{aligned} w^{\href{#eq-c1}{\textup{C1}}} &= \p{\theta_{1}^{\href{#eq-c1}{\textup{C1}}} - \rho \theta_{0}^{\href{#eq-c1}{\textup{C1}}} }^{\top}q + s, \end{aligned}\] (5.31)\[\begin{aligned} W^{\href{#eq-c2}{\textup{C2}}} &= P-Q, \end{aligned}\] (5.32)\[\begin{aligned} w^{\href{#eq-c2}{\textup{C2}}} &= p-q, \end{aligned}\] (5.33)\[\begin{aligned} W^{\href{#eq-c3}{\textup{C3}}} &= T - S, \end{aligned}\] (5.34)\[\begin{aligned} w^{\href{#eq-c3}{\textup{C3}}} &= t - s, \end{aligned}\] (5.35)\[\begin{aligned} W^{\href{#eq-c4}{\textup{C4}}} &= \p{\Theta_{1}^{\href{#eq-c4}{\textup{C4}}}}^{\top}S\Theta_{1}^{\href{#eq-c4}{\textup{C4}}} - \p{\Theta_{0}^{\href{#eq-c4}{\textup{C4}}}}^{\top}S\Theta_{0}^{\href{#eq-c4}{\textup{C4}}}, \end{aligned}\] (5.36)\[\begin{aligned} w^{\href{#eq-c4}{\textup{C4}}} &= \p{\theta_{1}^{\href{#eq-c4}{\textup{C4}}} - \theta_{0}^{\href{#eq-c4}{\textup{C4}}} }^{\top}s, \end{aligned}\] \[\begin{split}\begin{aligned} \PEPMaxIter_{\textup{cond}} &= \begin{cases} h + \alpha + 1& \text{ if } \textup{cond} \in \set{\href{#eq-c1}{\textup{C1}},\href{#eq-c3}{\textup{C3}}},\\ h & \text{ if } \textup{cond} \in \set{\href{#eq-c2}{\textup{C2}}}, \\ h + \alpha + 2 & \text{ if } \textup{cond} \in \set{\href{#eq-c4}{\textup{C4}}}, \end{cases} \nonumber \end{aligned}\end{split}\] and (5.37)\[\begin{split}\begin{aligned} \Theta_{0}^{\href{#eq-c1}{\textup{C1}}} &= \underbracket{ \begin{bmatrix} I_{n+\p{h+1}\NumEval } & 0_{\p{n+\p{h+1}\NumEval}\times\p{\alpha + 1}\NumEval } & 0_{\p{n+\p{h+1}\NumEval}\times m} \\ 0_{m \times \p{n+\p{h+1}\NumEval}} & 0_{m \times\p{\alpha + 1}\NumEval } & I_{m} \end{bmatrix} }_{ \in \reals^{\p{n+\p{h+1}\NumEval+m}\times\p{n+\p{h + \alpha +2}\NumEval+m}} }, \end{aligned}\end{split}\] (5.38)\[\begin{split}\begin{aligned} \theta_{0}^{\href{#eq-c1}{\textup{C1}}} &= \underbracket{ \begin{bmatrix} I_{\p{h+1}\NumEvalFunc} & 0_{\p{h+1}\NumEvalFunc \times \p{\alpha + 1}\NumEvalFunc } & 0_{\p{h+1}\NumEvalFunc \times \NumFunc } \\ 0_{\NumFunc\times \p{h+1}\NumEvalFunc} &0_{\NumFunc\times \p{\alpha + 1}\NumEvalFunc} & I_{\NumFunc} \end{bmatrix} }_{ \in \reals^{\p{\p{h+1}\NumEvalFunc + \NumFunc}\times\p{\p{h+\alpha+2}\NumEvalFunc + \NumFunc}} }, \end{aligned}\end{split}\] (5.39)\[\begin{split}\begin{aligned} \Theta_{1}^{\href{#eq-c1}{\textup{C1}}} &= \underbracket{ \begin{bmatrix} X_{\alpha + 1}^{0,h + \alpha + 1} \\ 0_{\p{\p{h + 1}\NumEval + m} \times \p{n + \p{\alpha + 1}\NumEval } } \quad I_{\p{h + 1}\NumEval + m} \end{bmatrix} }_{ \in \reals^{\p{n+\p{h+1}\NumEval+m} \times \p{n+\p{h + \alpha +2}\NumEval+m}} } , \end{aligned}\end{split}\] (5.40)\[\begin{aligned} \theta_{1}^{\href{#eq-c1}{\textup{C1}}} &= \underbracket{ \begin{bmatrix} 0_{\p{\p{h+1}\NumEvalFunc + \NumFunc} \times \p{\alpha+1}\NumEvalFunc } & I_{\p{h+1}\NumEvalFunc + \NumFunc } \end{bmatrix} }_{ \in \reals^{\p{\p{h+1}\NumEvalFunc + \NumFunc}\times\p{\p{h+\alpha+2}\NumEvalFunc + \NumFunc}} }, \end{aligned}\] (5.41)\[\begin{split}\begin{aligned} \Theta_{0}^{\href{#eq-c4}{\textup{C4}}} &= \underbracket{ \begin{bmatrix} I_{n+\p{h+\alpha+2}\NumEval } & 0_{\p{n+\p{h+\alpha+2}\NumEval}\times \NumEval } & 0_{\p{n+\p{h+\alpha+2}\NumEval}\times m} \\ 0_{m \times \p{n+\p{h+\alpha+2}\NumEval}} & 0_{m \times \NumEval } & I_{m} \end{bmatrix} }_{ \in \reals^{\p{n+\p{h+\alpha+2}\NumEval+m}\times\p{n+\p{h + \alpha +3}\NumEval+m}} }, \end{aligned}\end{split}\] (5.42)\[\begin{split}\begin{aligned} \theta_{0}^{\href{#eq-c4}{\textup{C4}}} &= \underbracket{ \begin{bmatrix} I_{\p{h+\alpha+2}\NumEvalFunc} & 0_{\p{h+\alpha+2}\NumEvalFunc \times \NumEvalFunc } & 0_{\p{h+\alpha+2}\NumEvalFunc \times \NumFunc } \\ 0_{\NumFunc\times \p{h+\alpha+2}\NumEvalFunc} &0_{\NumFunc\times \NumEvalFunc} & I_{\NumFunc} \end{bmatrix} }_{ \in \reals^{\p{\p{h+\alpha+2}\NumEvalFunc + \NumFunc}\times\p{\p{h+\alpha+3}\NumEvalFunc + \NumFunc}} }, \end{aligned}\end{split}\] (5.43)\[\begin{split}\begin{aligned} \Theta_{1}^{\href{#eq-c4}{\textup{C4}}} &= \underbracket{ \begin{bmatrix} X_{1}^{0,h + \alpha + 2} \\ 0_{\p{\p{h + \alpha + 2}\NumEval + m} \times \p{n + \NumEval } } \quad I_{\p{h + \alpha + 2}\NumEval + m} \end{bmatrix} }_{ \in \reals^{\p{n+\p{h+\alpha+2}\NumEval+m} \times \p{n+\p{h + \alpha +3}\NumEval+m}} } , \end{aligned}\end{split}\] (5.44)\[\begin{aligned} \theta_{1}^{\href{#eq-c4}{\textup{C4}}} &= \underbracket{ \begin{bmatrix} 0_{\p{\p{h+\alpha+2}\NumEvalFunc + \NumFunc} \times \NumEvalFunc } & I_{\p{h+\alpha+2}\NumEvalFunc + \NumFunc } \end{bmatrix} }_{ \in \reals^{\p{\p{h+\alpha+2}\NumEvalFunc + \NumFunc}\times\p{\p{h+\alpha+3}\NumEvalFunc + \NumFunc}} }. \end{aligned}\] Furthermore, if the interpolation conditions for \(\p{\mathcal{F}_{i}}_{i\in\IndexFunc}\) and \(\p{\mathcal{G}_{i}}_{i\in\IndexOp}\) are tight, \[\begin{split}\begin{aligned} & \textbf{for each}\ \textup{cond} \in\set{\href{#eq-c1}{\textup{C1}},\href{#eq-c2}{\textup{C2}},\href{#eq-c3}{\textup{C3}},\href{#eq-c4}{\textup{C4}}} \\ & \quad \dim \calH \geq n + \p{\PEPMaxIter_{\textup{cond}}+1}\NumEval + m, \\ & \mathbf{end} \end{aligned}\end{split}\] and there exists \[\begin{split}\begin{aligned} & \textbf{for each}\ \textup{cond} \in\set{\href{#eq-c1}{\textup{C1}},\href{#eq-c2}{\textup{C2}},\href{#eq-c3}{\textup{C3}},\href{#eq-c4}{\textup{C4}}} \\ & \quad G_{\textup{cond}}\in\sym_{++}^{n + \p{\PEPMaxIter_{\textup{cond}}+1}\NumEval + m}, \\ & \quad \bchi_{\textup{cond}}\in\reals^{\p{\PEPMaxIter_{\textup{cond}}+1}\NumEvalFunc + \NumFunc}, \\ & \mathbf{end} \end{aligned}\end{split}\] such that \[\begin{split}\begin{aligned} & \textbf{for each}\ \textup{cond} \in\set{\href{#eq-c1}{\textup{C1}},\href{#eq-c2}{\textup{C2}},\href{#eq-c3}{\textup{C3}},\href{#eq-c4}{\textup{C4}}} \\ & \quad \left( \begin{array}{@{}c@{}} \forall i \in\IndexFunc \\ \forall o \in \mathcal{O}^{\textup{func-ineq}}_{i} \\ \forall \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} \end{array} \right) \quad \begin{aligned} & \bm{\chi}^{\top}_{\textup{cond}} F_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-ineq}} \\ & + \trace\p{W_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-ineq}}G_{\textup{cond}}} \leq 0, \end{aligned} \\ & \quad \left( \begin{array}{@{}c@{}} \forall i \in\IndexFunc \\ \forall o \in \mathcal{O}^{\textup{func-eq}}_{i} \\ \forall \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} \end{array} \right) \quad \begin{aligned} & \bm{\chi}^{\top}_{\textup{cond}} F_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-eq}} \\ & + \trace\p{W_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{func-eq}}G_{\textup{cond}}} = 0, \end{aligned} \\ & \quad \left( \begin{array}{@{}c@{}} \forall i \in\IndexOp \\ \forall o \in \mathcal{O}^{\textup{op}}_{i} \\ \forall \p{ \p{j_{1},k_{1}}, \ldots, \p{j_{n_{i,o}},k_{n_{i,o}} }}\in\mathcal{J}_{i,o}^{0,\PEPMaxIter_{\textup{cond}}} \end{array} \right) \quad \trace\p{W_{\p{i,j_{1},k_{1},\ldots,j_{n_{i,o}},k_{n_{i,o}},o}}^{0,\,\PEPMaxIter_{\textup{cond}},\,\textup{op}}G_{\textup{cond}}} \leq 0, \\ & \mathbf{end} \end{aligned}\end{split}\] then [(5.28)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition) is also a necessary condition. *Proof.* By induction, we only need to consider the case \(k=0\) in [(C1)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c1) to [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4). Additionally, note that \[\begin{split}\begin{aligned} \p{\bx^{0},\bu^{0},\ldots,\bu^{h},\hat{\bu}^{\star},y^{\star}} &= \p{\Theta_{0}^{\href{#eq-c1}{\textup{C1}}} \kron \Id }\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}, \\ \p{\bFcn^{0},\ldots,\bFcn^{h},\bFcn^{\star}} &= \theta_{0}^{\href{#eq-c1}{\textup{C1}}}\p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}}, \\ \p{\bx^{\alpha + 1},\bu^{\alpha + 1},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}} &= \p{\Theta_{1}^{\href{#eq-c1}{\textup{C1}}} \kron \Id }\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}, \\ \p{\bFcn^{\alpha + 1},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} &= \theta_{1}^{\href{#eq-c1}{\textup{C1}}}\p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}}, \\ \p{\bx^{0},\bu^{0},\ldots,\bu^{h+\alpha + 1},\hat{\bu}^{\star},y^{\star}} &= \p{\Theta_{0}^{\href{#eq-c4}{\textup{C4}}} \kron \Id }\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 2},\hat{\bu}^{\star},y^{\star}}, \\ \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} &= \theta_{0}^{\href{#eq-c4}{\textup{C4}}} \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 2},\bFcn^{\star}}, \\ \p{\bx^{1},\bu^{1},\ldots,\bu^{h + \alpha + 2},\hat{\bu}^{\star},y^{\star}} &= \p{\Theta_{1}^{\href{#eq-c4}{\textup{C4}}} \kron \Id }\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 2},\hat{\bu}^{\star},y^{\star}}, \\ \p{\bFcn^{1},\ldots,\bFcn^{h + \alpha + 2},\bFcn^{\star}} &= \theta_{1}^{\href{#eq-c4}{\textup{C4}}} \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 2},\bFcn^{\star}}, \end{aligned}\end{split}\] where we have used [(5.37)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta0-c1-mat), [(5.38)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta0-c1), [(5.39)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta1-c1-mat), [(5.40)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta1-c1), [(5.41)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta0-c4-mat), [(5.42)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta0-c4), [(5.43)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta1-c4-mat), and [(5.44)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-theta1-c4), respectively. First, suppose that the parameters \(\p{Q,q,S,s}\) are fixed. Note that (5.45)\[\begin{split}\begin{aligned} &\mathcal{V}\p{Q,q,\alpha+1} - \rho \mathcal{V}\p{Q,q,0} + \mathcal{R}\p{S,s,0} \nonumber \\ & = \quadform{Q}{\p{\bx^{\alpha + 1},\bu^{\alpha + 1},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} + q^{\top}\p{\bFcn^{\alpha + 1},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} \nonumber \\ &\quad - \rho \quadform{Q}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h},\hat{\bu}^{\star},y^{\star}}} - \rho q^{\top}\p{\bFcn^{0},\ldots,\bFcn^{h},\bFcn^{\star}} \nonumber \\ &\quad + \quadform{S}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} + s^{\top}\p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} \nonumber \\ & = \quadform{\p{\Theta_{1}^{\href{#eq-c1}{\textup{C1}}}}^{\top}Q\Theta_{1}^{\href{#eq-c1}{\textup{C1}}} - \rho \p{\Theta_{0}^{\href{#eq-c1}{\textup{C1}}}}^{\top}Q\Theta_{0}^{\href{#eq-c1}{\textup{C1}}} + S}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} \nonumber \\ &\quad + \p{q^{\top}\p{\theta_{1}^{\href{#eq-c1}{\textup{C1}}} - \rho \theta_{0}^{\href{#eq-c1}{\textup{C1}}} } + s^{\top}} \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} \nonumber \\ & = \quadform{W^{\href{#eq-c1}{\textup{C1}}}}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} + \p{w^{\href{#eq-c1}{\textup{C1}}}}^{\top}\p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} \nonumber \end{aligned}\end{split}\] where [(5.29)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c1-mat) and [(5.30)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c1) are used in the last equality. Therefore, using [(5.45)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-obj-c1) as the objective function in [(PEP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#eq-pep), [Theorem 5.1.1 (Performance estimation via SDP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#thm-pep-leq-zero) gives that [(5.28)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition), with \(\textup{cond} = \href{#eq-c1}{\textup{C1}}\), is a sufficient condition for [(C1)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c1). Note that (5.46)\[\begin{split}\begin{aligned} &\mathcal{V}\p{P,p,0} - \mathcal{V}\p{Q,q,0} \notag \\ & = \quadform{P-Q}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h},\hat{\bu}^{\star},y^{\star}}} + \p{p-q}^{\top}\p{\bFcn^{0},\ldots,\bFcn^{h},\bFcn^{\star}} \notag \\ & = \quadform{W^{\href{#eq-c2}{\textup{C2}}}}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h},\hat{\bu}^{\star},y^{\star}}} + \p{w^{\href{#eq-c2}{\textup{C2}}}}^{\top}\p{\bFcn^{0},\ldots,\bFcn^{h},\bFcn^{\star}} \end{aligned}\end{split}\] where [(5.31)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c2-mat) and [(5.32)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c2) are used in the last equality. Therefore, using [(5.46)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-obj-c2) as the objective function in [(PEP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#eq-pep), [Theorem 5.1.1 (Performance estimation via SDP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#thm-pep-leq-zero) gives that [(5.28)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition), with \(\textup{cond} = \href{#eq-c2}{\textup{C2}}\), is a sufficient condition for [(C2)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c2). Note that (5.47)\[\begin{split}\begin{aligned} & \mathcal{R}\p{T,t,0} - \mathcal{R}\p{S,s,0} \notag \\ & = \quadform{T-S}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} + \p{t-s}^{\top}\p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} \notag \\ & = \quadform{W^{\href{#eq-c3}{\textup{C3}}}}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} + \p{w^{\href{#eq-c3}{\textup{C3}}}}^{\top} \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} \end{aligned}\end{split}\] where [(5.33)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c3-mat) and [(5.34)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c3) are used in the last equality. Therefore, using [(5.47)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-obj-c3) as the objective function in [(PEP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#eq-pep), [Theorem 5.1.1 (Performance estimation via SDP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#thm-pep-leq-zero) gives that [(5.28)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition), with \(\textup{cond} = \href{#eq-c3}{\textup{C3}}\), is a sufficient condition for [(C3)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c3). Note that (5.48)\[\begin{split}\begin{aligned} &\mathcal{R}\p{S,s,1} - \mathcal{R}\p{S,s,0} \notag \\ & = \quadform{S}{\p{\bx^{1},\bu^{1},\ldots,\bu^{h + \alpha + 2},\hat{\bu}^{\star},y^{\star}}} + s^{\top}\p{\bFcn^{1},\ldots,\bFcn^{h + \alpha + 2},\bFcn^{\star}} \notag \\ & \quad - \quadform{S}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 1},\hat{\bu}^{\star},y^{\star}}} - s^{\top} \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 1},\bFcn^{\star}} \notag \\ & = \quadform{\p{\Theta_{1}^{\href{#eq-c4}{\textup{C4}}}}^{\top}S\Theta_{1}^{\href{#eq-c4}{\textup{C4}}} - \p{\Theta_{0}^{\href{#eq-c4}{\textup{C4}}}}^{\top}S\Theta_{0}^{\href{#eq-c4}{\textup{C4}}}}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 2},\hat{\bu}^{\star},y^{\star}}} \notag \\ & \quad + \p{s^{\top}\p{\theta_{1}^{\href{#eq-c4}{\textup{C4}}} - \theta_{0}^{\href{#eq-c4}{\textup{C4}}} } } \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 2},\bFcn^{\star}} \nonumber \\ & = \quadform{W^{\href{#eq-c4}{\textup{C4}}}}{\p{\bx^{0},\bu^{0},\ldots,\bu^{h + \alpha + 2},\hat{\bu}^{\star},y^{\star}}} + \p{w^{\href{#eq-c4}{\textup{C4}}}}^{\top} \p{\bFcn^{0},\ldots,\bFcn^{h + \alpha + 2},\bFcn^{\star}} \end{aligned}\end{split}\] where [(5.35)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c4-mat) and [(5.36)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-w-c4) are used in the last equality. Therefore, using [(5.48)](https://autolyap.github.io/theory/iteration_independent_analyses/#equation-eq-iteration-independent-lyapunov-obj-c4) as the objective function in [(PEP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#eq-pep), [Theorem 5.1.1 (Performance estimation via SDP)](https://autolyap.github.io/theory/performance_estimation_via_sdps/#thm-pep-leq-zero) gives that [(5.28)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition), with \(\textup{cond} = \href{#eq-c4}{\textup{C4}}\), is a sufficient condition for [(C4)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-c4). Second, note that the proof is complete if we let the parameters \(\p{Q,q,S,s}\) free, as in [(5.28f)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition-q-mat), [(5.28g)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition-q), [(5.28h)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition-s-mat), and [(5.28i)](https://autolyap.github.io/theory/iteration_independent_analyses/#eq-iteration-independent-lyapunov-condition-s). \(\square\) TeX macro definitions used in this page: ```latex \newcommand{\Id}{\operatorname*{Id}} \newcommand{\IndexFunc}{\mathcal{I}_{\textup{func}}} \newcommand{\IndexOp}{\mathcal{I}_{\textup{op}}} \newcommand{\NumEval}{\bar{m}} \newcommand{\NumEvalFunc}{\bar{m}_{\textup{func}}} \newcommand{\NumFunc}{m_{\textup{func}}} \newcommand{\PEPMaxIter}{\bar{k}} \newcommand{\bFcn}{\mathbf{F}} \newcommand{\bchi}{\boldsymbol{\chi}} \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\calH}{\mathcal{H}} \newcommand{\kron}{\otimes} \newcommand{\llbracket}{\lbrack\!\lbrack} \newcommand{\naturals}{\mathbb{N}_{0}} \newcommand{\p}[1]{\mathord{( #1 )}} \newcommand{\quadform}[2]{\mathcal{Q}\p{#1,#2}} \newcommand{\reals}{\mathbb{R}} \newcommand{\rrbracket}{\rbrack\!\rbrack} \newcommand{\set}[1]{\mathord{\left.\{ #1 \} \right. }} \newcommand{\sym}{\mathbb{S}} \newcommand{\trace}{\mathrm{trace}} \newcommand{\underbracket}[1]{\underbrace{#1}} ```