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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

Hindi

सबसे बुरी स्थिति के सदस्यों के बारे में संभावित संस्करण के चार संस्करण प्रदर्शित करते हैं जहां अनुकूलन फ्रेम एल्गोरिथ्म असफल होते हैं: सिम्प्लेक्स विधि के लिए क्लेई-मिंटी संस्करण, खराब परिस्थितियों के लिए खराब परिस्थितियों और खराब परिस्थितियों के लिए स्क्रैपिंग बॉल, न्यूटन के विधि के लिए उन्नत मिनीमा के बारे में पूछते हैं, अपने स्वयं के उपकरणों के साथ स्थानीय क्लेई-मिंटी क्यूबों के लिए खराब संस्करण दिखाते हैं। यह सर्वेक्षण उन्हें एक साथ समीक्षा करता है। हम सात प्रतिनिधि एल्गोरिथ्म वर्गों के माध्यम से नियम विशिष्ट मामलों के इतिहास को ट्रैक करते हैं और उन्हें एक पढ़ने के साथ व्यव

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.

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

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