constraint modeling

To address the challenge of rigorously enforcing output constraints in safety-critical machine learning applications, this paper proposes a novel hyperspherical constraint representation method. It maps model outputs into a hyperspherical coordinate space centered at the feasible region, thereby intrinsically guaranteeing constraint satisfaction at the representation level—without penalty terms, custom architectures, or post-hoc projection. The approach uniformly supports both convex bounded and star-shaped feasible sets and provides theoretical guarantees of zero constraint violation. Key technical components include a hyperspherical coordinate transformation, geometry-driven feasible set modeling, constraint-aware feature mapping, and a lightweight inverse transformation. Experiments on synthetic and real-world datasets demonstrate that the method achieves prediction accuracy competitive with state-of-the-art constrained learning approaches, incurs no optimization overhead during inference, and strictly maintains zero constraint violations.

This work addresses the challenge of nonlinear optimization with mixed equality and inequality constraints in robotic dynamics planning by introducing a novel approach based on “constraint manifolds with corners.” The method reformulates the original problem as an unconstrained optimization over a constrained state space, seamlessly embedding inequality constraints into the manifold structure through differential geometry and manifold optimization techniques. This formulation overcomes the conventional limitation of manifold optimization, which typically applies only to smooth equality constraints. Evaluated on large-scale dynamic planning tasks, the proposed approach successfully generates dynamically feasible trajectories and demonstrates superior robustness and solvability in scenarios where standard algorithms fail.

Neural networks often struggle to strictly satisfy nonlinear constraints during inference, which hinders their deployment in safety-critical applications. This work proposes HardNet++, the first method capable of enforcing hard satisfaction of general nonlinear equality and inequality constraints, overcoming the limitation of existing approaches that are restricted to specific constraint forms. By integrating damped local linearization, differentiable projection layers, and end-to-end training, HardNet++ guarantees constraint compliance simultaneously during both training and inference. Evaluated on model predictive control tasks, HardNet++ achieves high-precision constraint adherence while preserving solution optimality.

This work addresses the lack of formal guarantees regarding semantic preservation during problem reformulation and solver correctness in constraint programming. It presents the first end-to-end verified framework implemented in the Lean theorem prover, enabling formal proofs of parameterized equivalence, equisatisfiability, and symmetry-breaking correctness for entire families of problems. The approach combines general, parameterized proofs with instance-level certificate checking, thereby eliminating the need to trust external solvers. Verified certificates are produced via backend transformations, and a single high-level proof suffices for arbitrarily large instances. This methodology achieves dramatic search-space reductions—up to a factor of twenty million—and enables full verification of the largest instances in just a few minutes.

This work addresses the problem of solving quantified constraints over the reals, involving existentially quantified parameters (user-specified) and universally quantified parameters (representing unknown or infinitely many scenario configurations). To overcome limitations in modeling expressiveness, systematic taxonomy, and cross-domain adaptability of existing approaches, we propose—firstly—a principled application classification scheme and literature indexing framework tailored to real-valued quantified constraints. Methodologically, we integrate Satisfiability Modulo Theories (SMT) over real arithmetic, quantifier elimination, interval analysis, and symbolic computation to enhance solver robustness and scalability. The contributions include an open, domain-spanning literature repository covering mathematics, control theory, formal verification, and optimization; standardized problem templates; and a publicly available benchmark suite. These advances significantly facilitate the engineering deployment of quantified reasoning tools within formal methods.

This work addresses the lack of systematic methodologies in model optimization, which often relies on heuristic choices and struggles to accommodate diverse deployment constraints. It formalizes model compression and acceleration as a constraint-aware multi-objective engineering decision problem, establishing a unified and actionable framework grounded in five key dimensions: data availability, latency, memory footprint, accuracy tolerance, and retraining budget. By integrating techniques such as quantization, pruning, knowledge distillation, parameter-efficient fine-tuning (PEFT), and inference optimization, the study proposes tailored optimization pipelines for four representative industrial scenarios, delivering a reproducible and quantifiable guide for technology selection.

This work addresses the critical limitation of existing multi-objective optimization methods—such as weighted sum, NNC, and NBI—which frequently fail to fully capture the Pareto front in problems with three or more objectives, often exhibiting omission rates exceeding 50%. To overcome this, the authors propose the Generalized Normal Constraint (GNC) method, which establishes a unified geometric and computational framework. By integrating Pareto front mesh construction with a normal constraint mechanism, GNC structurally guarantees 100% coverage of the feasible Pareto region for any n-objective problem. This approach fundamentally resolves the factorial degradation in coverage that plagues conventional techniques as the number of objectives increases. Theoretically ensuring solution set completeness, GNC significantly outperforms current methods and offers a more comprehensive solution for multi-objective optimization in fields such as engineering design and economic decision-making.

This work addresses the challenge of characterizing how constraints shape the geometric structure of feasible regions in constrained optimization. The authors propose a geometric framework grounded in self-adjoint operators, which constructs local reachable subspaces by encoding computational or feasibility constraints and yields a pseudoinverse-weighted gradient as the optimal first-order update direction. This approach unifies the treatment of projection, spectral truncation, and multi-objective feasibility, revealing a constraint-induced warped ascent geometry. It establishes a compatibility principle between spectral compression and multi-objective structures, with the algorithm dynamically focusing on dominant spectral modes to provide a unified geometric characterization of gradient projection, spectral compression, and multi-objective feasible directions.

This work addresses the lack of machine-verifiable formalizations of line search methods in nonlinear optimization, which has hindered algorithmic reliability. Within the Lean 4 theorem prover, it presents the first systematic formalization of several classical line search criteria—including Armijo, Goldstein, Wolfe, and their nonmonotone variants—alongside rigorous definitions of gradient descent, descent directions, and backtracking step-size selection. The study fully verifies the Zoutendijk convergence theorem within this framework, thereby establishing a comprehensive formal foundation for line search theory. This contribution significantly enhances the verifiability and trustworthiness of nonlinear optimization algorithms through mechanized mathematical reasoning.

This work addresses the hallucination problem in large language models when applied to precision-sensitive tasks such as technical drawing, where strict geometric constraints are often violated. To mitigate this, the authors propose PyGeoX, a programmable, geometry-specific domain language that compiles natural language descriptions into exact geometric constructions satisfying multiple constraints. By introducing a differentiable constraint loss and a saturation-based additive reward (SAR) mechanism that decomposes rewards per constraint, the approach effectively circumvents the issue of anomalous constraints masking gradients under global norm-based rewards, thereby preserving learning signals from partial progress. Experiments demonstrate that SAR improves solution success rates by 2.3× over an MSE-reward baseline on challenging tasks, and the resulting 8B-parameter model achieves performance on the newly introduced hierarchical benchmark PyGeoX-Bench that rivals substantially larger state-of-the-art systems.