A novel switched systems approach to nonconvex optimisation

We develop a novel switching dynamics that converges to the Karush-Kuhn-Tucker (KKT) point of a nonlinear optimisation problem. This new approach is particularly notable for its lower dimensionality compared to conventional primal-dual dynamics, as it focuses exclusively on estimating the primal variable. Our method is successfully illustrated on general quadratic optimisation problems, the minimisation of the classical Rosenbrock function, and a nonconvex optimisation problem stemming from the control of energy-efficient buildings.

💡 Research Summary

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The paper introduces a continuous‑time switched‑system algorithm for solving a class of non‑convex constrained optimization problems. Unlike traditional primal‑dual dynamics, which evolve both primal variables and Lagrange multipliers, the proposed method evolves only the primal vector z of dimension n, thereby reducing the state dimension from n + m + p to n. The key idea is to embed the inequality constraints directly into the dynamics through a switching law that determines which constraints are active at any instant.

Problem formulation: The authors consider an optimization problem
 min f(z)
 subject to A_eq(z)∇f(z)+d_eq = 0 (m equality constraints)
      A_inq(z)∇f(z)+d_inq ≤ 0 (p inequality constraints).
Here A_eq(z) and A_inq(z) are full‑rank matrices that depend on z, and the gradients of the constraints are expressed in terms of the gradient of the objective. This unconventional representation is chosen to simplify the later switched‑system design and can capture quadratic programming and several practical non‑convex problems.

KKT conditions are recalled as the target equilibrium: a point z* together with multipliers λ* and ν* must satisfy the usual stationarity, primal feasibility, dual feasibility, and complementary slackness conditions. Because the problem is non‑convex, these conditions are only necessary; a second‑order sufficient condition involving the Hessian of the Lagrangian is also presented.

Switched‑system background: A family of vector fields h_s(z) indexed by a finite set S and a piecewise‑constant switching signal σ(t) define the dynamics ẋ = h_{σ(t)}(x). The authors adopt Carathéodory solutions, an average dwell‑time condition to avoid Zeno behavior, and the concepts of a common Lyapunov function and weak invariance.

Algorithm construction:

1. Active‑set identification – At any state z, the set of constraints with g_inq_i(z) ≥ 0 is declared active.
2. Subsystem definition – For each possible active set 𝒜, a subsystem is defined as
    h_𝒜(z) = −∇f(z) − A_eq(z)^T λ_𝒜 − A_inq(z)^T ν_𝒜,
   where the multipliers for active inequalities are implicitly set to one (or any positive value) and those for inactive constraints are zero. This yields a vector field that mimics the stationarity condition of the KKT system without explicitly computing multipliers.
3. Switching law – The switching signal σ(t) = 𝒜(z(t)) updates whenever a constraint crosses zero, thereby ensuring that the feasible set Γ = G_eq ∩ G_inq remains positively invariant, similar to interior‑point barrier methods.

Stability analysis: The objective function f(z) itself is used as a common Lyapunov function V(z) = f(z) − f(z*). For any subsystem h_𝒜, the authors prove ∇V·h_𝒜 ≤ 0, guaranteeing non‑increase of V along trajectories. By imposing an average dwell‑time bound on the switching, they ensure that V strictly decreases unless the state is already at a KKT point. Moreover, the Lyapunov function satisfies an Input‑to‑State Stability (ISS) inequality, providing robustness to disturbances and to variations in the initial condition.

Convergence proof: Under the technical assumptions that (i) ∇f is radially unbounded, (ii) A_eq(z) and A_inq(z) are full rank, and (iii) the switching satisfies the average dwell‑time condition, the authors show that every complete trajectory remains bounded, stays inside Γ, and its limit set consists of points satisfying the KKT conditions. Hence the algorithm converges (in the sense of Lyapunov) to a KKT point of the original non‑convex problem.

Numerical experiments: Three case studies illustrate the method’s practicality.

• Quadratic programming – The switched dynamics converge to the optimal solution with a state dimension equal to the number of decision variables, outperforming a comparable primal‑dual ODE that requires additional dual states.
• Constrained Rosenbrock function – Despite the notorious narrow valley of the Rosenbrock landscape, the algorithm finds the global minimum while respecting the inequality constraints, whereas standard gradient descent gets trapped in local minima.
• Energy‑efficient building control – The method is applied to the steady‑state optimization of HVAC operating points, a problem featuring nonlinear equality and inequality constraints derived from thermodynamic balances. The switched system successfully identifies the energy‑optimal operating point and demonstrates real‑time applicability.

The paper concludes by highlighting the advantages of the switched‑system formulation: reduced dimensionality, ISS‑based robustness, and a large domain of attraction equal to the feasible set G_inq. Limitations include the need for radially unbounded gradients, full‑rank constraint Jacobians, and the requirement to compute the switching law online, which may be computationally demanding for very large‑scale problems. Future work is suggested on relaxing these assumptions, extending the approach to distributed optimization over networks, and exploring adaptive dwell‑time strategies.