触发：当问题涉及资源分配、取舍、最大化/最小化目标，或在限制条件下做决策，或需要判断问题的凸性、使用拉格朗日方法、分析对偶结构时调用。 模式：科研模式适用于数学推导、论文审查、实验设计优化；生活模式适用于日常决策、时间管理、人际互动、生活规划。 Trigger when a problem involves resource allocation, trade-offs, maximizing/minimizing objectives, decision-making under constraints, or when convexity analysis, Lagrangian methods, or duality structures are relevant. Mode: Research mode for mathematical derivation, paper review, experimental design optimization; Life mode for daily decisions, time management, interpersonal interactions, life planning.