Newest papers

U-Examples: A Unified Family of Worst-Case Instances for Optimization Algorithms

Eghbal Hosseini · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.27909 · cited by 0 works (OpenAlex) · open-access PDF

Korean

최악의 경우 구성원에 대 한 동일한 복잡한 복합 요소는 최적화 프레임 알고리즘이 실패하는 곳을 보여줍니다 : Klee-Minty instance 큐브는 간단한 방법에 대 한, 나쁜 조건과 분리 된 맵에 대 한 성능 추락, 뉴턴의 방법에 대 한 동일한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한 복잡한

Machine translation (Meta M2M100); the English original follows.

Abstract

Worst-case instances show where optimization algorithms fail: the Klee-Minty cube for the simplex method, ill-conditioned and rippled bowls for gradient descent, degenerate minima for Newton's method, redundant Klee-Minty cubes for interior-point methods, unprunable trees for branch-and-bound, deceptive landscapes for metaheuristics, and degenerate cycling for active-set methods. These constructions were developed over seventy years in separate literatures, each with its own tools. This survey reviews them together. We trace the history of rule-specific worst cases across seven representative algorithm classes and organize them with one reading: in each, the optimum lies near the start, yet the algorithm's own update rule is misdirected onto a long trajectory or, at the boundary, into a non-terminating loop. We call such an instance a U-Example; the letter U stands both for the unification and for the U-shaped bowls around which descent methods are made to crawl. A common framework treats every algorithm as a rule on a state space with a progress potential, so that one definition of misdirection, and of the resulting blow-up in cost, covers pivoting, descent, interior-point, tree-search, and population methods. Each member is drawn in a common visual grammar and quantified in general dimension $n$. The blow-up is independent of the dimension for the two descent members and exponential in it for four search members, with degenerate cycling as a breakdown. We connect the constructions to neighbouring theories (oracle complexity, performance estimation, smoothed analysis, and the complexity of local search), each of which asks a different question of the same instances. We then map each worst case to the remedy that defeats it, examine how robust the instances are, and close with open challenges, in particular whether a worst case can be predicted from the update rule alone.

Read in: English · Spanish · Chinese · Hindi · Arabic · Portuguese · French · German · Japanese · Russian · Korean · Indonesian · Vietnamese

Topics: Dimension (graph theory), Descent (aeronautics), Algorithm, Computer science, Degenerate energy levels, Population

Sources: OpenAlex (CC0), arXiv, Crossref. Agent Exchange