发表机构
Clemson University（克莱姆森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对无约束规范时二阶锥规划的对偶间隙问题，本文仅用洛伦兹锥约束构造精确对偶，并证明其可行性与对偶值可达性，回答了Pólik-Terlaky问题。
AI 中文摘要
在没有约束规范的情况下，二阶锥规划可能出现对偶间隙或对偶值不可达。对于每个具有任意实系数的此类规划，我们仅使用仿射方程和洛伦兹锥乘积中的隶属关系构造了一个精确对偶。每个可行的对偶点都给出一个有效界。只要原始问题可行且具有有限值，对偶就能达到该值。这回答了Pólik和Terlaky提出并在后续工作中重申的问题：二阶锥规划的精确对偶是否只能使用洛伦兹锥约束。同样的构造产生一个仿射二阶锥系统，当给定的仿射二阶锥系统不可行时（包括弱不可行情形），该系统恰好可行。两个公式都具有多项式大小，并且仅使用固定的有理常数从输入系数统一获得。这些是关于精确实数数据的公式大小结果；它们并不意味着多项式时间可解性或有理证书位长度的多项式界。
英文摘要
Without a constraint qualification, second-order cone programs can have a duality gap or an unattained dual value. For every such program with arbitrary real coefficients, we construct an exact dual using only affine equations and memberships in products of Lorentz cones. Every feasible dual point gives a valid bound. Whenever the primal is feasible with finite value, the dual attains that value. This answers a question raised by Pólik and Terlaky and reiterated in subsequent work: whether an exact dual for second-order cone programming can use only Lorentz-cone constraints. The same construction produces an affine second-order cone system that is feasible exactly when a given affine second-order cone system is infeasible, including in the weakly infeasible case. Both formulations have polynomial size and are obtained uniformly from the input coefficients using only fixed rational constants. These are formulation-size results over exact real data; they do not imply polynomial-time solvability or polynomial bounds on the bit length of rational certificates.
