Linearly Constrained DC Programming

• Linearly constrained DC programming is an optimization framework that decomposes a difference-of-convex objective while enforcing feasibility via linear constraints.
• It applies methods such as DCA, SCA, and augmented Lagrangian smoothing to solve nonconvex problems in quadratic, polynomial, and combinatorial contexts.
• Recent studies show its efficiency through scalable algorithms that achieve faster convergence and have practical applications in resource allocation and statistical inference.

Linearly constrained DC programming studies optimization problems where the objective is a difference of convex (DC) functions and the feasible set is defined by linear or affine constraints. Such problems appear in nonconvex quadratic programming, polynomial optimization, combinatorial relaxations, and statistical inference. Contemporary research focuses on developing convergent, scalable algorithms for these inherently nonconvex tasks, exploiting the DC structure and the tractability of linear constraints.

1. Problem Formulation and Theoretical Foundations

A standard linearly constrained DC program is formulated as

$$\min_{x\in\mathbb{R}^n} \; F(x) := g(x) - h(x) \quad \text{subject to}\quad A x = b,$$

where $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$ are closed, convex (often with additional differentiability properties), and $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$. The feasible set may further include explicit convex sets or polyhedral constraints, as in

$$X = \{ x\in\mathbb{R}^n : A x = b,\; C x \leq d \}.$$

Stationarity concepts are derived from the DC structure: a point $x^*$ is stationary if there exists $\lambda\in\mathbb{R}^m$ such that

$$0 \in \partial g(x^*) - \partial h(x^*) + A^\top \lambda,\quad A x^* = b,$$

where $\partial h$ denotes a suitable subdifferential depending on smoothness. $\epsilon$-approximate stationarity typically means the primal, dual, and DC 'gaps' are all at most $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$0.

2. Principal Algorithmic Paradigms

Various algorithmic frameworks have been developed for linearly constrained DC programs:

• Difference-of-Convex Algorithm (DCA) and Boosted DCA (BDCA): At each iteration, a convex subproblem is formed by linearizing the concave term $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$1 at the current point and minimizing the resulting convex approximation subject to the constraints. BDCA incorporates an extrapolation/line-search phase to accelerate convergence by exploiting sufficient decrease conditions. The update scheme can be written as $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$2 for $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$3, with $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$4 chosen via line search along $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$5 to ensure a decrease in the DC objective (Artacho et al., 2019, Zhang et al., 2022).
• Successive Convex Approximation (SCA): For block-structured problems, SCA linearizes both the nonseparable components off the diagonal and the concave parts, adding proximal regularization, to create a strongly convex surrogate. This surrogate is optimized subject to the linear (and possibly separable coupling) constraints, often enabling decomposition across agents or variables. Dual or primal decomposition leverages the linear structure for distributed optimization (Alvarado et al., 2013).
• Augmented Lagrangian with Moreau-Envelope Smoothing: The recent LCDC-ALM framework applies a difference-of-Moreau-envelopes (DME) smoothing to both components of the DC objective, yielding a smooth surrogate. This is combined with an augmented Lagrangian to enforce affine constraints, resulting in algorithms that attain an $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$6 iteration complexity for $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$7-stationary solutions (Sun et al., 2021).
• Cutting Plane Algorithms for Mixed-Binary Programs: DC programs arising from exact penalty reformulations of mixed-integer problems are addressed by combining DCA optimization of the continuous penalty relaxation with sequential addition of DC-based cuts (affine underestimators of the integrality measure $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$8), yielding globally valid inequalities and global convergence with the interplay of combinatorial and convex optimization (Niu et al., 2021).

3. Specialized Decompositions and Structural Exploitation

For problem classes such as linearly constrained polynomial programs, the construction of a DC decomposition plays a central role:

• Power-Sum DC Decompositions: Any polynomial $g,h: \mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$9 (especially even-degree or via homogenization) is decomposed as $A\in\mathbb{R}^{m\times n}$0 where $A\in\mathbb{R}^{m\times n}$1 are sums of even powers of affine functions, determined by solving a sparse linear system. This enables the usage of accelerated DC optimization methods while maintaining explicit exploitation of the polynomial and constraint structure (Zhang et al., 2022).
• Block-Separability and Distributed Methods: For multi-agent or resource allocation problems with block variables $A\in\mathbb{R}^{m\times n}$2 and shared affine or linear constraints (e.g., $A\in\mathbb{R}^{m\times n}$3), the SCA or BDCA formulations naturally decouple across blocks, supporting parallel or distributed computation. Dual updates are handled via projected subgradient ascent on the Lagrange multipliers, with blockwise primal subproblems solved efficiently (Alvarado et al., 2013).
• Indicator Functions and Polyhedral Constraints: Linear or polyhedral constraints are incorporated directly via indicator functions in the convex part of the DC decomposition (e.g., $A\in\mathbb{R}^{m\times n}$4), preserving convex subproblem tractability and yielding simple KKT systems at DC-critical points (Artacho et al., 2019).

4. Convergence Properties and Complexity Results

Major algorithmic schemes for linearly constrained DC programs possess rigorous convergence guarantees:

• DCA/BDCA: Under standard coercivity and Slater-type assumptions, all cluster points of BDCA iterates satisfy the KKT conditions for the original DC program. For quadratic objectives, global R-linear convergence holds, i.e., $A\in\mathbb{R}^{m\times n}$5 (Artacho et al., 2019).
• SCA: With strongly convex surrogate construction and appropriately chosen or diminishing step sizes, every limit point is a stationary solution. Inexact subproblem solves are permitted, provided error terms diminish sufficiently fast ($A\in\mathbb{R}^{m\times n}$6) (Alvarado et al., 2013).
• LCDC-ALM: For both smooth and composite models, an $A\in\mathbb{R}^{m\times n}$7-stationary point is obtained in $A\in\mathbb{R}^{m\times n}$8 iterations. The smoothing technique ensures level-boundedness and allows for nonsmooth concave terms in the objective, extending applicability relative to standard ALM approaches for weakly convex optimization (Sun et al., 2021).
• Cutting Plane with DCA Restarts: If every fractional vertex of the LP relaxation eventually yields a global DC or classical (Lift-and-Project) cut, and cuts are not repeatedly generated, lower bounds converge to the optimum and the gap achieves $A\in\mathbb{R}^{m\times n}$9-optimality within finitely many iterations (Niu et al., 2021).

5. Practical Implementations and Numerical Performance

Recent studies extensively benchmark the performance of modern linearly constrained DC programming algorithms:

• High-dimensional Quadratic and Trust-Region Problems: BDCA achieves significant speedups (3–15×) over classical DCA across copositivity detection, $b\in\mathbb{R}^m$0 and $b\in\mathbb{R}^m$1 trust-region subproblems, and piecewise-quadratic box-constrained problems, with boosting via line search activated in 40–80% of iterations (Artacho et al., 2019).
• Polynomial Optimization: For MVSK portfolio models and high-dimensional box-constrained polynomial minimization, power-sum DC decomposition with BDCA equipped with exact line search (BDCAₑ) outperforms existing DCA, Armijo-type BDCA, and standard nonlinear solvers (FMINCON, FILTERSD) as problem dimension or polynomial degree increases, especially for dense instances (Zhang et al., 2022).
• Resource Allocation in Communication and Signal Processing: Distributed SCA algorithms efficiently solve nonconvex sum-rate maximization in MIMO cognitive radio networks and secure multi-user resource games, often matching or surpassing centralized solutions in both objective value and convergence speed (Alvarado et al., 2013).
• Mixed-Binary Linear Programming: DCCUT algorithms leveraging DC cuts outperform classical cutting-plane loops (LAPCUT), closing lower bounds more rapidly and finding optimal or near-optimal solutions across various test sets and MIPLIB 2017 instances, especially when integrated with parallel strategies (Niu et al., 2021).

6. Extensions and Ongoing Directions

Ongoing research addresses enhanced smooth approximations (e.g., difference-of-Moreau-envelopes), general nonlinear or semidefinite constraints, and further acceleration mechanisms—including inertial terms and KL-based global convergence rates. The power-sum DC decomposition provides a systematic template for higher-order nonconvex polynomial objectives. In mixed-integer optimization, DC-based penalization and cutting-plane methods are increasingly integrated with parallelism and classical combinatorial optimization (Sun et al., 2021, Zhang et al., 2022, Niu et al., 2021).

Emerging trends also include distributed methods for networked and multi-agent nonconvex optimization, as well as primal-dual and block-coordinate variants that exploit problem separability for scalable computation on modern hardware.

7. Summary Table: Key Methods and Complexity

Algorithmic Scheme	Best-Case Complexity or Rate	Applicability Domain
BDCA / DCA	R-linear (quadratic case), descent	Smooth DC with polyhedral linear constraints (Artacho et al., 2019)
SCA (distributed)	Stationary-point global convergence	Block-structured, linearly coupled DC (Alvarado et al., 2013)
LCDC-ALM	$b\in\mathbb{R}^m$2 iteration bound	General linearly constrained DC, possible nonsmooth (Sun et al., 2021)
DCCUT (with DCA restarts)	Finite convergence to $b\in\mathbb{R}^m$3-optimality	Mixed-binary LP/DC with DC cuts (Niu et al., 2021)
BDCAe (exact line search + PSDC)	Sublinear / linear (KL property)	High-degree polynomials, explicit affine constraints (Zhang et al., 2022)

These results underscore the feasibility of solving large-scale nonconvex optimization problems with linear constraints by exploiting DC structure and tailored decomposition, smoothing, and cut-generation techniques.