AI 中文总结
研究将参数平均收益博弈框架扩展到双目标热带伪线性优化，通过参数博弈刻画帕累托前沿，给出联合分母界等结果，开发方向二分和牛顿方案两种算法，并进行数值实验验证算法复杂度。
AI 中文摘要
我们将Parsons等人的参数平均收益博弈框架扩展到具有一般双边约束的双目标热带伪线性优化。问题是在一般双边系统\(U\otimes x\oplus b\leq V\otimes x\oplus d\)的可行集上同时最小化两个热带伪线性目标。我们通过两个参数\((\lambda_1,\lambda_2)\)的参数平均收益博弈来刻画帕累托前沿。可行性区域\(R\)是凸的，帕累托前沿\(P\)是具有有限多个断点的凸分段线性曲线。我们给出了联合分母界这一新结果，通过参数博弈的循环结构给出了最优性和不可行性证书。开发了两种算法，方向二分算法和牛顿方案，并进行了数值实验。
英文摘要
We extend the parametric mean-payoff game framework of Parsons et al. to bi-objective tropical pseudolinear optimization with general two-sided constraints. The problem is to simultaneously minimize two tropical pseudolinear objectives over the feasible set of a general two-sided system U otimes x oplus b is less than or equal to V otimes x oplus d, we characterize the Pareto front via a parametric mean-payoff game in two parameters ( lambda 1, lambda 2). The feasibility region R is convex and the Pareto front P is a convex piecewise-linear curve with finitely many breakpoints, these properties are natural extensions of the single-parameter case to two parameters. In addition, we give as a new result, the joint denominator bound: the cycle coefficients satisfy k 1( gamma ) + k 2( gamma ) is less than or equal to 2 for any elementary cycle gamma, yielding | Delta | is less than or equal to 2 for every 2 times 2 Newton system, except in the fully decoupled case, and implying that all breakpoints have half-integer coordinates for integer data. Optimality and infeasibility certificates are given in terms of the cycle structure of the parametric game. Two algorithms are developed, a directional bisection algorithm (O(n squared (n+m) log M) per direction) and a Newton scheme tracing the complete Pareto front via 2 times 2 linear solves in at most | S | steps, independent of M. The directional bisection algorithm is pseudo-polynomial in n, m and M. The Newton scheme is independent of M but requires up to |S| steps, where |S| is exponential in n. Lastly, we give numerical experiments on random instances to confirm the directional bisection complexity bound exactly; the Newton scheme's worst-case bound is not attained by random instances but is shown to be tight via explicit adversarial constructions.
Comments38 pages, 8 figures
