Lossless Convexification

• Lossless Convexification is a framework that reformulates non-convex problems into convex ones via variable transformations and convex hull constructions.
• It guarantees global or near-global optimality by ensuring conditions like surjectivity, Slater’s condition, and normality are satisfied.
• Its applications span semidefinite programming, mixed-integer programming, and learning with discrete losses, enabling robust and scalable optimization.

Lossless convexification is a mathematical framework by which certain non-convex optimization or control problems are transformed, via change of variables or convex hull constructions, into convex programs whose optimal solutions coincide—under precise structural conditions—with those of the original problem. This equivalence can be global (no loss) or nearly global (with bounded violations determined by system dimensions or grid resolution). The framework appears in semidefinite programming, optimal control, mixed-integer quadratic and conic optimization, learning with discrete structures, and combinatorial surrogate loss design.

1. Structural Definition and Core Conditions

The canonical definition involves replacing a non-convex feasible set or cost by a “convexification” such that (a) all original feasible points map to the convex set, and (b) every feasible point of the convexified problem has a preimage in the non-convex set (“surjectivity condition”).

For an optimization problem

$$\min_{x \in \mathbb{R}^n} \; f(x) \quad \text{s.t.} \quad \Phi_i(x) \preceq 0, \quad i=1, ..., N,$$

where $f$ and the matrix-valued constraints are non-convex, a lossless convexification exists if there is a variable change $v = h(x)$ and convex $f', \Phi_i'$ such that

$f(x) = f'(h(x)), \quad \Phi_i(x) = \Phi_i'(h(x)),$

and every feasible $v$ in the convexified problem is $v = h(x)$ for some feasible $x$ (Lee, 2021). This ensures both problems share the same optimal value and, possibly modulo preimage multiplicity, optimal points.

2. Semidefinite Programming and Strong Duality

In semidefinite programming, Lee (Lee, 2021) rigorously establishes that lossless convexification preserves strong duality for non-convex SDPs under Slater’s condition. The original non-convex program

$$\min_{x \in \mathbb{R}^n} f(x) \quad \text{s.t.} \quad \Phi_i(x) \preceq 0$$

may be non-convex in $f$ and/or in the constraint maps. After convex change-of-variable, the convexified SDP enjoys zero duality gap provided the surjectivity condition holds and there exists a strictly feasible $f$0 (all $f$1). This enables full dual KKT characterizations and often Riccati-equation representations in control (Lee, 2021).

3. Lossless Convexification in Optimal Control and Mixed-Integer Programming

Many non-convex and mixed-integer optimal control problems admit lossless convexifications via convex epigraphical relaxations in continuous time, and under certain dimensionality constraints, in discretized time.

3.1 Semi-continuous Inputs and SOCP

For actuator selection with inputs constrained to be either $f$2 (inactive) or norm-bounded ($f$3), the convex relaxation replaces integer logic variables $f$4 by $f$5 and introduces slack variables $f$6 with second-order cone (SOC) constraints (Malyuta et al., 2019, Malyuta et al., 2019). If the system satisfies observability, normality, and transversality (“no abnormal multiplier”) conditions, then the SOCP solution recovers bang-bang controls $f$7 and norm constraints at solution, yielding global optimality (Malyuta et al., 2019, Malyuta et al., 2019).

3.2 Discrete-Time and DLCvx

In discrete-time, lossless convexification is more delicate: exactness may hold at all grid points only up to $f$8 violations, where $f$9 is the state-space dimension. Perturbation analysis and normality characterization show that, after an arbitrarily small generic perturbation to the recursion matrix, at most $v = h(x)$0 grid points may fail to satisfy the original non-convex constraint (e.g., control norm lower bound) (Luo et al., 2024, Kiami, 2024). For first-order-hold parameterizations, the bound increases to $v = h(x)$1 vertices and $v = h(x)$2 edges. For “long-horizon” problems, a value-function bisection restores near-lossless properties (Luo et al., 2024, Kiami, 2024).

For discrete-time pointing and mixed-integer sector constraints, a dual multiplier–based analysis bounds possible slack violations, and the convex relaxation remains globally tight modulo a small, quantifiable number of grid anomalies (Luo et al., 12 Jan 2025).

3.3 Discrete-Valued and Fuel-Optimal Control

In fuel-optimal control with discrete actuator profiles, lossless convexification by convexification to a (cross-)polytopic relaxation (e.g., $v = h(x)$3-ball) preserves the extremal/bang-bang property provided a unique Hamiltonian minimization holds (adjoint vector normality). All convexified solutions automatically saturate at the true (discrete) control values (Arenas-Uribe et al., 11 Nov 2025).

4. Lossless Convexification in Quadratic and Perspective Reformulation

The lossless convexification principle underlies the “perspective reformulation” for epigraphs of mixed-integer quadratic or convex programs involving step or semicontinuous penalties. The convex hull of a set $v = h(x)$4 is

$v = h(x)$5

where $v = h(x)$6 is convex ("perspective function") (Lee et al., 2020). This convex envelope strictly dominates product and “big-M” relaxations and is SOCP-representable for $v = h(x)$7 (Lee et al., 2020).

For quadratic programs with finite-valued step-function penalties (e.g., SVM with $v = h(x)$8-$v = h(x)$9 loss), rank-one convexification yields ideal SDP or copositive relaxations, recovers all integral optima, and is robust to label noise (Choi et al., 23 Apr 2025).

5. Convexification of Discrete Losses and Learning from Constraints

In empirical risk minimization with non-modular, discrete losses (e.g., Dice or multi-label losses), lossless convexification is achieved via unique submodular-supermodular decompositions and tight convex surrogates (e.g., Lovász-hinge for submodular part, slack-rescaling for supermodular part) (Yu et al., 2016). The resulting convex surrogate is piecewise-linear, matches the discrete loss at the “vertices” (i.e., hard labels), and is polynomial-time subgradient computable—lossless convexification in the sense of extension to the original discrete problem (Yu et al., 2016).

In learning from constraints with MIP structure and complex loss/regularization, the tightest convex extension (Legendre-Fenchel biconjugate) is lossless on the vertices; efficient per-instance decompositions permit scalable computation and preserve exact minima with fractional relaxations (Shcherbatyi et al., 2016).

6. Functionals with Structural Penalties: Convex Envelope

For objectives of the form $f', \Phi_i'$0, the lower semi-continuous convex envelope can be computed losslessly via the $f', \Phi_i'$1-transform when $f', \Phi_i'$2 is identity (or under spectral majorization/minorization otherwise) (Carlsson, 2016). The condition $f', \Phi_i'$3 certifies global optimality and lossless recovery of structured minimizers (sparse, low rank) with no relaxation gap. This construction generalizes $f', \Phi_i'$4 or nuclear norm relaxations and bridges to the convex hull of step-function and rank constraints.

7. Unified Perspective and Applications

Lossless convexification is a structurally sensitive paradigm: its global optimality hinges on observability/normality/Slater-type conditions, surjectivity of variable change, or combinatorial geometric properties. It enables polynomial-time, robust global optimization in settings previously considered intractable, including mixed-integer optimal control, quadratic programming with indicator variables, and learning under hard combinatorial constraints.

Key properties:

Domain	Structural Condition	Nature of Guarantee
Non-convex SDP	Surjective change of variables, Slater	Zero duality gap, KKT necessary/sufficient
Mixed-integer OC	Observability/normality, SOC relaxation	Global optima almost everywhere
Perspective/MINLO	Convex perspective, epigraph tightness	Convex hull, exact SOCP
Discrete surrogate	Extension property, unique decomposition	Lossless surrogate, polynomial-time computation
General structured $f', \Phi_i'$5	$f', \Phi_i'$6-transform, spectrum comparison	Envelope losslessness at minimizers

The approach unifies and subsumes classical relaxations, offering sharp a priori or a posteriori certificates of losslessness, and informs the design of scalable algorithms for non-convex and mixed-integer problems across control, learning, and combinatorial optimization (Lee, 2021, Yu et al., 2016, Malyuta et al., 2019, Choi et al., 23 Apr 2025, Luo et al., 2024, Arenas-Uribe et al., 11 Nov 2025, Carlsson, 2016).