无需降维至ℝⁿ→ℝ，优化ℝⁿ→ℝⁿ映射的技术问询

Great question—framing it with the energy/momentum physics analogy really helps ground the idea of optimizing multiple, potentially conflicting outputs directly. The short answer is yes, there are well-established frameworks that handle vector-valued optimization without first collapsing it to a scalar objective. Here are the key approaches you should explore:

• Pareto Multi-Objective Optimization
  This is the most common framework for your scenario. Instead of forcing a single "best" solution by weighting outputs into a scalar (like summing energy and momentum with arbitrary coefficients), Pareto optimization finds the Pareto frontier: a set of solutions where you can't improve one output without worsening another. For your energy/momentum example, every point on this frontier represents a meaningful tradeoff—say, higher energy at the cost of lower momentum, or vice versa. Algorithms like NSGA-II or MOEA/D operate directly on the vector output space, searching for this frontier without scalarization. They evaluate solutions based on dominance (whether one solution outperforms another across all outputs) rather than a single scalar score.

• Lexicographic Optimization
  If you have a clear priority order for your outputs (e.g., "maximize energy first, then momentum given that energy is as large as possible"), lexicographic ordering lets you optimize directly in vector space. It works by first solving for the optimal value of the highest-priority output, then restricting the feasible region to solutions that achieve that optimal value, and repeating for the next priority. This avoids scalarization entirely—you're essentially optimizing a sequence of nested problems, each focused on one component of the output vector.

• Set-Valued Optimization
  This is a more generalized framework where the "solution" to the optimization problem is a set of outputs rather than a single point. It directly models the mapping from input space to sets of optimal outputs, which is perfect for problems where multiple conflicting outputs can't be reduced to a scalar. Set-valued optimization uses concepts from convex analysis and functional analysis to characterize these solution sets, working entirely within the vector space without scalar conversion.

• Geometric Vector Optimization
  For convex vector optimization problems (where the feasible region and objective functions have convex structure), you can use geometric tools to directly analyze the optimal solutions. For example, the Pareto frontier of a convex problem is often a convex subset of the output space, and you can use separation theorems to find points on this frontier without scalarizing. This approach leans heavily on the geometry of ℝⁿ, treating the output vector as a cohesive geometric object rather than a collection of disconnected scalars.

A quick note on the "mapping to scalar" approach you mentioned: while it's common (because scalar optimization tools are highly developed), it's not the only way. The methods above all operate directly on the vector output, respecting the multidimensional nature of your problem. Your energy/momentum analogy is particularly apt here—classically, these are linked via physical laws, but if you're in a scenario where you're trying to optimize both (e.g., designing a propulsion system), Pareto optimization would give you the full range of feasible tradeoffs without forcing an arbitrary weight between the two.

内容的提问来源于stack exchange，提问作者Cort Ammon