Sequential Convex Programming Methods for Solving Nonlinear Optimization Problems with DC constraints

This paper investigates the relation between sequential convex programming (SCP) as, e.g., defined in [24] and DC (difference of two convex functions) programming. We first present an SCP algorithm for solving nonlinear optimization problems with DC constraints and prove its convergence. Then we combine the proposed algorithm with a relaxation technique to handle inconsistent linearizations. Numerical tests are performed to investigate the behaviour of the class of algorithms.

💡 Research Summary

The paper investigates the connection between sequential convex programming (SCP) and difference‑of‑convex (DC) programming for solving nonlinear optimization problems that contain DC‑type constraints. The authors consider a problem (P) where the objective f : ℝⁿ→ℝ is convex, the feasible set Ω⊂ℝⁿ is closed and convex, and each constraint component g_i : ℝⁿ→ℝ can be written as a difference of two convex functions, g_i = u_i − v_i. The set of feasible points is D = { x∈Ω | g(x)≤0 }.

The main contribution is an SCP algorithm specifically tailored to DC constraints, called SCP‑DC. At iteration k a subgradient matrix Ξ_k∈∂v(x_k) is computed (∂v denotes the subdifferential of the convex part v). Using a first‑order linearization of v, the original nonconvex constraint g(x)≤0 is replaced by the convex surrogate

 u(x) − v(x_k) − Ξ_k (x − x_k) ≤ 0.

Consequently the subproblem

 (P(x_k)) min f(x) s.t. u(x) − v(x_k) − Ξ_k (x − x_k) ≤ 0, x∈Ω

is a convex program that can be solved globally (e.g., by interior‑point methods). Under a Slater‑type interior‑point condition the solution of (P(x_k)) satisfies the KKT conditions of the subproblem. The algorithm proceeds by repeatedly (1) computing Ξ_k, (2) solving (P(x_k)) to obtain x_{k+1} and its multiplier λ_{k+1}, and (3) checking a stopping criterion based on ‖x_{k+1}−x_k‖.

Theoretical analysis shows that, if the original problem satisfies Clarke’s calmness constraint qualification, the generated sequence converges to a KKT point of (P). Moreover, when the objective and the convex components are strongly convex (ρ_f>0), the convergence is locally linear. The paper also discusses the importance of a good DC decomposition: a decomposition with small strong‑convexity parameters leads to easier subproblems and faster convergence.

A practical difficulty arises when the linearization of the concave part v yields an infeasible subproblem. To address this, the authors introduce a relaxation technique based on an exact L₁‑penalty. The relaxed subproblem adds the term

 θ ‖max{0, u(x) − v(x_k) − Ξ_k (x − x_k)}‖₁

to the objective, where θ>0 is a penalty weight. This guarantees feasibility of the subproblem even when the original linearization is inconsistent, and it reduces the conservatism typical of penalty‑based DC methods. The relaxed algorithm (SCP‑DC‑R) therefore allows full step sizes without line‑search or trust‑region safeguards.

Two motivating examples illustrate the methodology. The first is a nonlinear model predictive control (NMPC) problem for a bilinear dynamical system. The bilinear dynamics are expressed as quadratic forms, each of which is decomposed into a difference of two positive‑semidefinite matrices, yielding DC constraints. The second example is a mathematical program with complementarity constraints (MPCC). By introducing a slack variable and using the identity 2 xᵀz = ‖(x,z)‖² − ‖x − z‖², the complementarity condition xᵀz = 0 is rewritten as a DC inequality.

Numerical experiments on both problems compare SCP‑DC, its relaxed variant, and traditional DC algorithms (DCA, CCCP). Results show that SCP‑DC converges in fewer iterations and with larger step sizes. In cases where the linearization of v is frequently infeasible, the relaxed version remains stable and reaches comparable objective values, whereas standard penalty‑based DC methods suffer from excessive conservatism and small steps.

In summary, the paper provides a rigorous bridge between SCP and DC programming, proposes a practically implementable SCP‑DC algorithm with provable convergence, and enhances robustness through an L₁‑penalty relaxation. The work extends the applicability of DC‑based methods to higher‑dimensional, structured nonconvex problems commonly encountered in control, engineering design, and complementarity‑type optimization.