Anytime Solvers for Variational Inequalities:
the (Recursive) Safe Monotone Flows

The study of the interplay between continuous-time dynamical systems and variational inequalities has a rich history. In the context of optimization, classical references include  [2, 3, 4, 5]. For general variational inequalities, flows solving them can be obtained through differential inclusions involving monotone set-valued maps, originally introduced in [6]. These systems have been equivalently described as a projected dynamical systems [7] and complementarity systems [8, 9]. Recently, this framework has been extended to settings where monotonicity holds with respect to non-Euclidean norms [10], and when the system evolves on a Riemannian manifold [11]. One limitation of these systems is that they are, in general, discontinuous, which poses challenges both for their theoretical analysis and practical implementation.

The interconnection of algorithms with physical plants has attracted much attention recently. The interconnected system can often be viewed as the coupling of a dynamical system with a set valued operator (cf. [12, 13] for a comprehensive survey of systems having this form). The specific setting where the algorithm optimizes the steady-state of the plant is typically referred to as online feedback optimization[14, 15] and has been studied in continuous time in the context of power systems [16, 17], network congestion control [18], and traffic networks [19], as well as discrete time in [20]. Recently this framework has also been studied in game scenarios [21, 22].

A complementary approach uses extremum seeking control [23], which has been generalized to the setting of games in [24]. Extremum seeking differs from the methods introduced here in that they are typically zeroth-order methods, and do not offer exact stability guarantees. The recent work [25] considers safety guarantees for extremum seeking control for the special case of a single inequality constraint. In fact, the proposed algorithm can be understood as an approximation of the safe gradient flow in [26], which is a precursor for constrained optimization of the algorithms proposed here for constraint sets parameterized by multiple inequality and equalities.

To synthesize our flows, we employ techniques from safety-critical control [27, 28], which refers to the problem of designing a feedback controller that ensures that the state of the system satisfies certain constraints. The problem of ensuring safety is typically formalized by specifying a set of states where the system is said to remain safe, and ensuring the safe set is forward invariant. The work [29] reviews set invariance in control. A popular technique for synthesizing safe controllers uses the concept of control barrier functions (CBFs), see [30, 31, 32] and references therein, to specify optimization-based feedback controllers which “filter” a nominal system to ensure it remains in the safe set. Here, we apply this strategy to synthesize anytime algorithms, viewing the constraint set as a safety set and the monotone operator of the variational inequality as the nominal system. This view has connections to projected dynamical systems, whose relationship with CBF-based control design has recently been explored in [33, 34], and leads to the alternative “projection-based” interpretation of the projected monotone flow and safe monotone flow proposed here.

We consider the synthesis of continuous-time dynamical systems solving variational inequalities while respecting the constraints at all times. We work primarily with continuous-time systems since we are motivated by online feedback optimization, where the evolution of a physical process determines the functions that define the variational inequality, which in turn regulates the dynamically evolving plant. The methods introduced here can be implemented on digital hardware by discretizing the resulting dynamics.

We discuss three flows that solve this problem. The projected monotone flow is already known, but we provide a novel control-theoretic interpretation as a control system whose closed-loop behavior is as close as possible to the monotone operator while still belonging to the tangent cone of the constraint set. The safe monotone flow can be interpreted either as a control system with a feedback controller synthesized using techniques from safety-critical control or as an approximation of the projected monotone flow. The latter interpretation relies on the novel notion of restricted tangent set, which generalizes the usual concept of tangent cone from variational geometry. We show that equilibria correspond exactly with critical points of the original problem, establish existence and uniqueness of solutions, and characterize the regularity properties of flow. We also show that the constraint set is forward invariant and asymptotically stable, and derive global stability guarantees under the additional assumption of convexity and monotonicity. Our technical analysis relies on a suite of Lyapunov functions to establish stability properties with respect to the constraint set and the whole state space. When the constraint set is polyhedral, we establish that the system is contracting and exponentially stable. Finally, the recursive safe monotone flow bypasses the need for continuously solving quadratic programs along the trajectories by incorporating a dynamics whose equilibria precisely correspond to such solutions, and interconnecting the dynamical systems on different time scales. Using tools from singular perturbation theory for contracting systems, we show that for variational inequalities with polyhedral constraints, the KKT points are locally exponentially stable and globally attracting, and obtain practical safety guarantees. We compare the three flows on a simple two-player game and demonstrate how the safe monotone flow can be interconnected with dynamical processes on an example of a receding horizon linear quadratic dynamic game.

The flows introduced here generalize the safe gradient flow, a continuous-time system proposed in our previous work [26] (in parallel, [20] introduced a discrete-time implementation of a simplified version of it). With respect to the safe gradient flow, our treatment extends the results in three key ways. First, we consider variational inequalities, rather than just constrained optimization problems, making the flows introduced here applicable to a much broader range of problems. Second, with assumptions of monotonicity and convexity of the constraints, we obtain global stability and convergence results, rather than local stability results. Third, the rigorous characterization of the contractivity properties of the safe monotone flow paves the way for its interconnection with other dynamically evolving processes. In fact, the proposed recursive safe monotone flow critically builds on this analysis by leveraging different timescales and singular perturbation theory.

Let denote the set of real numbers. For $c\in$, $[c]_{+}=\max\{0,c\}$. For $x\in^{n}$, $x_{i}$ denotes the $i$th component and $x_{-i}$ denotes all components of $x$ excluding $i$. For $v,w\in^{n}$, $v\leq w$ (resp. $v<w$) denotes $v_{i}\leq w_{i}$ (resp. $v_{i}<w_{i}$) for $i\in\{1,\dots,n\}$. We let $\left\lVert v\right\rVert$ denote the Euclidean norm. We write $Q\succeq 0$ (resp., $Q\succ 0$) to denote $Q$ is positive semidefinite (resp., $Q$ is positive definite). Given $Q\succeq 0$, let $\left\lVert x\right\rVert_{Q}$ denote the seminorm where $\left\lVert x\right\rVert_{Q}=\sqrt{x^{\top}Qx}$. For a symmetric $Q$, $\lambda_{\text{min}}(Q)$ and $\lambda_{\textnormal{max}}{(Q)}$ denote the minimum and maximum eigenvalues of $Q$, resp. Given $\mathcal{C}\subset^{n}$, the distance of $x\in^{n}$ to $\mathcal{C}$ is $\textnormal{dist}(x,\mathcal{C})=\inf_{y\in\mathcal{C}}\left\lVert x-y\right\rVert$. The projection map onto $\overline{\mathcal{C}}$ is $\text{proj}_{\mathcal{C}}:^{n}\rightrightarrows\overline{\mathcal{C}}$, where $\textnormal{proj}_{\mathcal{C}}\left(x\right)=\big{\{}y\in\overline{\mathcal{C}}\mid\left\lVert x-y\right\rVert=\textnormal{dist}(x,\mathcal{C})\big{\}}$. Given a closed and convex set $\mathcal{C}\subset^{n}$, the normal cone to $\mathcal{C}$ at $x\in^{n}$ is $N_{\mathcal{C}}(x)=\{d\in^{n}\mid d^{\top}(x^{\prime}-x)\leq 0,\;\forall x^{\prime}\in\mathcal{C}\}$ and the tangent cone to $\mathcal{C}$ at $x$ is $T_{\mathcal{C}}(x)=\{\xi\in^{n}\mid d^{\top}\xi\leq 0,\;\forall d\in N_{\mathcal{C}}(x)\}$.

Given $g:^{n}\to$, we denote its gradient by $\nabla g$. For $g:^{n}\to^{m}$, $\frac{\partial g(x)}{\partial x}$ denotes its Jacobian. For $I\subset\{1,2,\dots,m\}$, we denote by $\frac{\partial g_{I}(x)}{\partial x}$ the matrix whose rows are $\{\nabla g_{i}(x)^{\top}\}_{i\in I}$. Given a function $f:^{n}\to$, we say it is directionally differentiable if for all $\xi\in^{n}$, the following limit exists

$$f^{\prime}(x;\xi)=\lim_{h\to 0^{+}}\frac{f(x+h\xi)-f(x)}{h}.$$

Given a vector field $\mathcal{G}:^{n}\to^{n}$ and a function $V:^{n}\to$, the Upper-right Dini derivative of $V$ along $\mathcal{G}$ is

$$D^{+}_{\mathcal{G}}V(x)=\limsup_{h\to 0^{+}}\frac{1}{h}\left(V(\Phi_{h}(x))-V(x)\right),$$

where $\Phi_{h}$ is the flow map¹¹1 For $h\in_{\geq 0}$, the flow map $\Phi_{h}:^{n}\to^{n}$ is defined by $\Phi_{h}(x_{0})=x(h)$, where $x(t)$ is the unique trajectory solving $\dot{x}=\mathcal{G}(x)$ with $x(0)=x_{0}$. corresponding to the system $\dot{x}=\mathcal{G}(x)$. When $V$ is directionally differentiable $D^{+}_{\mathcal{G}}V(x)=V^{\prime}(x,\mathcal{G}(x))$ and when $V$ is differentiable, then $D^{+}_{\mathcal{G}}V(x)=\nabla V(x)^{\top}\mathcal{G}(x)$.

Here we review the basic theory of variational inequalities following [35]. Let $F:^{n}\to^{n}$ be a map and $\mathcal{C}\subset^{n}$ a set of constraints. A variational inequality refers to the problem of finding $x^{*}\in\mathcal{C}$ such that

$$(x-x^{*})^{\top}F(x^{*})\geq 0,\qquad\forall x\in\mathcal{C}.$$

(1)

We use $\textnormal{VI}(F,\mathcal{C})$ to refer to the problem (1) and $\textnormal{SOL}(F,\mathcal{C})$ to denote its set of solutions. Variational inequalities provide a framework to study many different analysis and optimization problems, including

• •

  Solving the nonlinear equation $F(x^{*})=0$, which corresponds to $\textnormal{VI}(F,^{n})$;

• •

  Minimizing the function $f:^{n}\to$ subject to the constraint that $x\in\mathcal{C}$, which corresponds to $\textnormal{VI}(\nabla f,\mathcal{C})$;

• •

  Finding saddle points of the function $\ell:^{n}\times^{m}\to$ subject to the constraints that $x_{1}\in X_{1}$ and $x_{2}\in X_{2}$, which corresponds to $\textnormal{VI}([\nabla_{x_{1}}\ell;-\nabla_{x_{2}}\ell],X_{1}\times X_{2})$.

• •

  Finding the Nash equilibria of a game with $N$ agents, where the $i$th agent wants to minimize the cost $J_{i}(x_{i},x_{-i})$ subject to the constraint $x_{i}\in X_{i}$, which corresponds to $\textnormal{VI}(F,\mathcal{C})$, where $F$ is the pseudogradient operator defined by $F(x)=(\nabla_{x_{1}}J_{1}(x),\dots,\nabla_{x_{N}}J_{N}(x))$, and $\mathcal{C}=X_{1}\times X_{2}\times\cdots\times X_{N}$.

The map $F:^{n}\to^{n}$ is monotone if $(x_{1}-x_{2})^{\top}(F(x_{1})-F(x_{2}))\geq 0$, for all $x_{1},x_{2}\in^{n}$, and $F$ is $\mu$-strongly monotone if there exists $\mu>0$ such that $(x_{1}-x_{2})^{\top}(F(x_{1})-F(x_{2}))\geq\mu\left\lVert x_{1}-x_{2}\right\rVert^{2}$, for all $x_{1},x_{2}\in^{n}$. When $F$ is a gradient map, i.e., $F=\nabla f$ for some function $f:^{n}\to$, then monotonicity (resp. $\mu$-strong monotonicity) is equivalent to convexity (resp. strong convexity) of $f$. When $F$ is monotone and $\mathcal{C}$ is convex, $\textnormal{VI}(F,\mathcal{C})$ is a monotone variational inequality.

In order to provide a characterization of the solution set $\textnormal{SOL}(F,\mathcal{C})$, we need to introduce a more explicit description of the set of constraints. Suppose that $g:^{n}\to^{m}$ and $h:^{n}\to^{k}$ are continuously differentiable and $\mathcal{C}$ is described by inequality constraints and affine equality constraints,

$$\mathcal{C}=\{x\in^{n}\mid g(x)\leq 0,\;h(x)=Hx-c_{h}=0\},$$

(2)

where $H\in^{k\times n}$ and $c_{h}\in^{k}$. For $x\in^{n}$, we denote the active constraint, inactive constraint, and constraint violation sets, resp., as

	$\displaystyle I_{0}(x)$	$\displaystyle=\{i\in[1,m]\mid g_{i}(x)=0\},$
	$\displaystyle I_{-}(x)$	$\displaystyle=\{i\in[1,m]\mid g_{i}(x)<0\},$
	$\displaystyle I_{+}(x)$	$\displaystyle=\{i\in[1,m]\mid g_{i}(x)>0\}.$

The problem (1) satisfies the constant-rank condition at $x\in\mathcal{C}$ if there exists an open set $U$ containing $x$ such that for all $I\subset I_{0}(x)$, the rank of $\{\nabla g_{i}(y)\}_{i\in I}\cup\{\nabla h_{j}(y)\}_{j=1}^{k}$ remains constant for all $y\in U$. The problem (1) satisfies the Mangasarian-Fromovitz Constraint Qualification (MFCQ) condition at $x\in\mathcal{C}$ if $\{\nabla h_{j}(x)\}_{i=1}^{k}$ are linearly independent, and there exists $\xi\in^{n}$ such that $\nabla g_{i}(x)^{\top}\xi<0$ for all $i\in I_{0}(x)$, and $\nabla h_{j}(x)^{\top}\xi=0$ for all $j\in\{1,\dots,k\}$. If MFCQ holds at $x^{*}\in\mathcal{C}$, then, if $x^{*}$ satisfies (1), there exists $(u^{*},v^{*})\in^{m}\times^{k}$ such that

	$\displaystyle F(x^{*})+\sum_{i=1}^{m}u^{*}_{i}\nabla g_{i}(x^{*})+\sum_{j=1}^{k}v_{j}^{*}\nabla h_{j}(x^{*})$	$\displaystyle=0$		(3a)
	$\displaystyle g(x^{*})$	$\displaystyle\leq 0$		(3b)
	$\displaystyle h(x^{*})$	$\displaystyle=0$		(3c)
	$\displaystyle u^{*}$	$\displaystyle\geq 0$		(3d)
	$\displaystyle(u^{*})^{\top}g(x^{*})$	$\displaystyle=0.$		(3e)

Equations (3) are called the KKT conditions. A point $(x^{*},u^{*},v^{*})$ satisfying them is a KKT triple and the pair $(u^{*},v^{*})$ is a Lagrange multiplier. For monotone variational inequalities, when MFCQ holds at $x^{*}$, then the KKT conditions are both necessary and sufficient for $x^{*}\in\textnormal{SOL}(F,\mathcal{C})$. When $F$ is monotone $\textnormal{SOL}(F,\mathcal{C})$ is closed and convex. If $F$ is additionally $\mu$-strongly monotone, then the set of solutions is a singleton.

We review here notions from the theory of set invariance for control systems following [29] and discuss methods for synthesizing feedback controllers that ensure it. Consider a control-affine system

	$\displaystyle\dot{x}$	$\displaystyle=\mathcal{F}(x,\nu)$		(4)
		$\displaystyle=F_{0}(x)+\sum_{i=1}^{r}\nu_{i}F_{i}(x),$

with Lipschitz-continuous vector fields $F_{i}:^{n}\to^{n}$, for $i\in\{0,\dots,r\}$, and a set $\mathcal{U}\subset^{r}$ of valid control inputs $\nu$. Let $\mathcal{C}\subset^{n}$ be a constraint set of the form (2) to which we want to restrict the evolution of the system. We consider the problem of designing a feedback controller $\kappa:^{n}\to\mathcal{U}$ such that $\mathcal{C}$ is forward invariant with respect to the closed-loop dynamics $\dot{x}=\mathcal{F}(x,\kappa(x))$. In applications, $\mathcal{C}$ often corresponds to the set of states for which the system can operate safely. For this reason, we refer to $\mathcal{C}$ as the safety set, and call the system safe under a controller $\kappa$ if $\mathcal{C}$ is forward invariant. A controller ensuring safety is safeguarding. We discuss two optimization-based strategies for synthesizing safeguarding controllers.

Our design strategy originates from the observation that, when $F$ is monotone, the system $\dot{x}=-F(x)$ finds solutions to the unconstrained variational inequality $\textnormal{VI}(F,^{n})$. However, trajectories flowing along this dynamics might leave the constraint set $\mathcal{C}$. This leads us to consider the control-affine system:

	$\displaystyle\dot{x}$	$\displaystyle=\mathcal{F}(x,u,v)$		(7)
		$\displaystyle=-F(x)-\sum_{i=1}^{m}u_{i}\nabla g_{i}(x)-\sum_{j=1}^{k}v_{j}\nabla h_{j}(x).$

Here, we have augmented the system with inputs from the admissible set $\mathcal{U}=^{m}_{\geq 0}\times^{k}$ to modify the flow of the original drift $-F$ to account for the constraints in a way that ensures that the solutions to (7) stay inside of or approach $\mathcal{C}$. The idea is that if the constraint $g_{i}(x)\leq 0$ is in danger of being violated, the corresponding input $u_{i}$ can be increased to ensure trajectories continue to satisfy it. Likewise, the input $v_{j}$ can be increased or decreased to ensure the corresponding constraint $h_{j}(x)=0$ is satisfied along trajectories.

Our design proceeds by thinking of $\mathcal{C}$ as a safety set for the system and using the approach outlined in Section 2.3.1 to synthesize a safeguarding feedback controller $(u,v)=\kappa(x)$. Assuming that MFCQ holds for all $x\in\mathcal{C}$, the map $K_{\text{proj}}:^{n}\rightrightarrows^{m}_{\geq 0}\times^{k}$ takes the form

	$\displaystyle K_{\text{proj}}(x)=\Big{\{}(u,v)\in^{m}_{\geq 0}\times^{k}\;\Big{|}$	$\displaystyle-\frac{\partial g_{I_{0}}}{\partial x}F(x)-\frac{\partial g_{I_{0}}}{\partial x}\frac{\partial g}{\partial x}^{\top}u-\frac{\partial g_{I_{0}}}{\partial x}\frac{\partial h}{\partial x}^{\top}v\leq 0,$		(8)
		$\displaystyle-\frac{\partial h}{\partial x}F(x)-\frac{\partial h}{\partial x}\frac{\partial g}{\partial x}^{\top}u-\frac{\partial h}{\partial x}\frac{\partial h}{\partial x}^{\top}v=0\Big{\}}.$

The following result states that the set of admissible controls is nonempty. We omit its proof for space reasons, but note that it readily follows from Farka’s Lemma [39].

We then use the feedback controller

$$\kappa(x)\in\underset{(u,v)\in K_{\text{proj}}(x)}{\text{argmin}}J(x,u,v),$$

(9)

where we set the objective function to be

$$J(x,u,v)=\frac{1}{2}\bigg{\|}\sum_{i=1}^{m}u_{i}\nabla g_{i}(x)+\sum_{j=1}^{k}v_{j}\nabla h_{j}(x)\bigg{\|}^{2}.$$

(10)

This function measures the magnitude of the “modification” of the drift term in (7). Thus, the QP-based controller (9) has the interpretation, at each $x$, of finding the control input such that the closed-loop system dynamics are as close as possible to $-F(x)$, while still being in $T_{\mathcal{C}}(x)$. In general, the program given by (9) does not have unique solutions. Despite this, we show below that the closed-loop dynamics of (7) is well defined regardless of which solution to (5) is chosen. We refer to it as the projected monotone flow.

The second implementation of the projected monotone flow consists of projecting $-F(x)$ onto the tangent cone of the constraint set. In general, the tangent cone does not have a representation that allows us to compute the projection easily. However, when the appropriate constraint qualification condition holds, the tangent cone admits a convenient parameterization which allows for the projection to be implemented as a quadratic program. Let $x\in\mathcal{C}$ and suppose that MFCQ holds at $x$. It follows that the tangent cone can be parameterized²²2 The right hand side of (11) is called the linearization cone of $\mathcal{C}$, and denoted as $T^{\text{lin}}_{\mathcal{C}}(x)$. In general, $T^{\text{lin}}_{\mathcal{C}}(x)\subset T_{\mathcal{C}}(x)$, and we say that Abadie’s Constraint Qualification (ACQ) [40] holds at $x\in\mathcal{C}$ if the reverse inclusion holds, i.e., $T^{\text{lin}}_{\mathcal{C}}(x)=T_{\mathcal{C}}(x)$. MFCQ is a stronger condition that implies ACQ. Many of the results we state in the paper can be generalized to hold under the weaker assumption of ACQ, however for simplicity, we avoid a detailed technical discussion of constraint qualifications. as

$\displaystyle T_{\mathcal{C}}(x)=\bigg{\{}\xi\in^{n}\;\Big{|}\;\frac{\partial h(x)}{\partial x}\xi=0,\frac{\partial g_{I_{0}}(x)}{\partial x}\xi\leq 0\bigg{\}}.$

(11)

The projection-based implementation of the projected monotone flow takes then the following form:

	$\displaystyle\dot{x}$	$\displaystyle=\textnormal{proj}_{T_{\mathcal{C}}(x)}\left(-F(x)\right)$		(12)
		$\displaystyle=\underset{\xi\in^{n}}{\text{argmin}}\quad\frac{1}{2}\left\lVert\xi+F(x)\right\rVert^{2}$
		$\displaystyle\quad\;\text{subject to}\quad\frac{\partial g_{I_{0}}(x)}{\partial x}\xi\leq 0,\frac{\partial h(x)}{\partial x}\xi=0.$

The projection onto the tangent ensures that $\mathcal{C}$ is forward invariant, by Nagumo’s Theorem [29, Theorem 3.1].

Here, we lay out the properties of the projected monotone flow. We begin by establishing the equivalence between the control- and projection-based implementations. We then discuss existence and uniqueness of solutions, and finally the stability and safety properties of the dynamics.

We start with the control system (7) with the admissible control set $\mathcal{U}=^{m}_{\geq 0}\times^{k}$, viewing $\mathcal{C}$ as a safety set, and design a safeguarding controller. We synthesize this controller using the function $(g,h)$ as a VCBF, following the approach outlined in Section 2.3.2.

Letting $\alpha>0$ be a parameter, the set of control inputs ensuring safety is given by

	$\displaystyle K_{\textnormal{cbf},\alpha}(x)=\Big{\{}(u,v)\in_{\geq 0}^{m}\times^{k}\;\Big{|}$	$\displaystyle-\frac{\partial g}{\partial x}F(x)-\frac{\partial g}{\partial x}\frac{\partial g}{\partial x}^{\top}u-\frac{\partial g}{\partial x}\frac{\partial h}{\partial x}^{\top}v\leq-\alpha g(x)$
		$\displaystyle-\frac{\partial h}{\partial x}F(x)-\frac{\partial h}{\partial x}\frac{\partial g}{\partial x}^{\top}u-\frac{\partial h}{\partial x}\frac{\partial h}{\partial x}^{\top}v=-\alpha h(x)\Big{\}}.$

The next result shows that this set is nonempty on an open set containing $\mathcal{C}$.

The proof of this result is identical to [26, Lemma 4.1] and we omit it for brevity. By Lemma 5.1, the feedback controller $(u,v)=\kappa(x)$ where

$$\kappa(x)\in\underset{(u,v)\in K_{\textnormal{cbf},\alpha}(x)}{\text{argmin}}J(x,u,v),$$

(13)

and $J$ is given by (10), is well defined on $X$. This controller has the same interpretation as before: determining the control input belonging to $K_{\textnormal{cbf},\alpha}(x)$ such that the closed-loop system dynamics are as close as possible to $-F(x)$. Similar to the case with projection-based methods, the problem (6) does not necessarily have unique solutions. However, we show below that the closed-loop system is well-defined regardless of which solution is chosen. We refer to it as the safe monotone flow with safety parameter $\alpha$, denoted $\dot{x}=\mathcal{G}_{\alpha}(x)$.

Here we describe the implementation of the safe monotone flow as a projected dynamical system. Similar to the projected monotone flow, the projected system is obtained by projecting $-F(x)$ onto a set-valued map. However, because the projection onto the tangent cone is in general discontinuous as a function of the state, we replace the tangent cone with the $\alpha$-restricted tangent set (here $\alpha>0$), denoted $T_{\mathcal{C}}^{(\alpha)}$, defined as

$\displaystyle T^{(\alpha)}_{\mathcal{C}}(x)=\Big{\{}\xi\in^{n}\;\Big{|}\;\frac{\partial g(x)}{\partial x}\xi\leq-\alpha g(x),\frac{\partial h(x)}{\partial x}\xi=-\alpha h(x)\Big{\}}.$

(14)

Figure 1 illustrates this definition. This set can be interpreted as an approximation of the usual tangent cone, but differs in several key ways. First, the restricted tangent set is not a cone, meaning that vectors in $T^{(\alpha)}_{\mathcal{C}}(x)$ cannot be scaled arbitrarily: in certain direction, the magnitude of vectors in $T^{(\alpha)}_{\mathcal{C}}(x)$ is restricted. An important property of $T^{(\alpha)}_{\mathcal{C}}(x)$ is that, even though the tangent cone is undefined for $x\not\in\mathcal{C}$, this is not the case for the restricted tangent set. In fact, it can be shown that $T_{\mathcal{C}}^{(\alpha)}$ takes nonempty values on an open set containing $\mathcal{C}$. This property allows for the safe monotone flow to be well-defined for infeasible initial conditions. The next result states properties of the $\alpha$-restricted tangent set.

Using the $\alpha$-restricted tangent set, we can define the projected dynamical system

	$\displaystyle\dot{x}$	$\displaystyle=\textnormal{proj}_{T^{(\alpha)}_{\mathcal{C}}(x)}\left(-F(x)\right)$		(15)
		$\displaystyle=\underset{\xi\in^{n}}{\text{argmin}}\quad\frac{1}{2}\left\lVert\xi+F(x)\right\rVert^{2}$
		$\displaystyle\quad\text{subject to}\quad\frac{\partial g(x)}{\partial x}\xi\leq-\alpha g(x)$
		$\displaystyle\quad\phantom{subjectto}\quad\frac{\partial h(x)}{\partial x}\xi=-\alpha h(x).$

Similar to the projected monotone flow, the projection operation ensures that the trajectories of the system remain in the safety set. However, as we show next, the advantages of projecting onto the restricted tangent cone is that the system is well defined for infeasible initial conditions, and trajectories of the system are smooth.

We now discuss the properties of the safe monotone flow. We begin by establishing the equivalence of the control-based and projection-based implementations. Next, we discuss its stability and safety properties.