[论文解读] Optimization Problems with Nearly Convex Objective Functions and Nearly Convex Constraint Sets

Nguyen Nang Thieu, Nguyen Dong Yen|arXiv (Cornell University)|Feb 10, 2026

Optimization and Variational Analysis被引用 0

一句话总结

该论文研究近凸目标与近凸约束集的优化问题，构造一个唯一确定的相关凸问题，其目标函数下半连续，推导了来自凸问题的极值条件与拉格朗日乘子规则。并比较了两种近凸集合的概念，并给出示例说明。

ABSTRACT

To every nearly convex optimization problem, that is a minimization problem with a nearly convex objective function and a nearly convex constraint set, we associate a uniquely defined convex optimization problem with a lower semicontinuous objective function and a closed constraint set. Interesting relationships between the original nearly convex problem and the associated convex problem are established. Optimality conditions in the form of Fermat's rules are obtained for both problems. We then get a Lagrange multiplier rule for a nearly convex optimization problem under a geometrical constraint and functional constraints from the Kuhn-Tucker conditions for the associated convex optimization problem. The obtained results are illustrated by concrete examples.

研究动机与目标

• Motivate and formalize nearly convex optimization problems where both the objective and constraints are nearly convex.
• Associate each nearly convex problem to a uniquely defined convex problem with a lower semicontinuous objective and closed constraint set.
• Derive Fermat-type optimality conditions for both the original and the associated convex problems.
• Obtain a Lagrange multiplier rule for nearly convex problems via Kuhn–Tucker conditions of the associated convex problem.
• Compare two notions of nearly convex sets (Minty vs Ho) and illustrate results with examples.

提出的方法

• Define nearly convex functions and sets, and establish properties of their domains, epigraphs, and normal cones.
• Construct the associated convex problem by replacing the original objective with its lower semicontinuous hull bar and closing the constraint set to bar.
• Prove equality of optimal values between the original and associated problems under a regularity condition ri(D) 2 ri(dom f) and ri(D) 2 ri(dom fbar).
• Derive Fermat-type optimality conditions for both problems using subdifferentials of nearly convex functions.
• Obtain a Lagrange multiplier rule for geometrical and functional constraints by invoking Kuhn-Tucker conditions for the associated convex problem.
• Discuss two notions of near convexity (Minty vs Ho) and how they influence optimality results.

实验结果

研究问题

• RQ1 When does the optimal value of a nearly convex optimization problem equal the optimal value of its associated convex problem?
• RQ2 What are the necessary and sufficient Fermat-type optimality conditions for nearly convex problems and their convex associates?
• RQ3 How can Lagrange multipliers be obtained for nearly convex problems via the associated convex problem's Kuhn-Tucker conditions?
• RQ4 How do two notions of nearly convex sets (Minty and Ho) differ in implications for optimality in nearly convex optimization?
• RQ5 Under what regularity conditions do the solution sets of the nearly convex problem and its convex associate relate (e.g., inclusion S S1)?

主要发现

• An associated convex problem can be constructed whose optimal value matches the original under ri(D) 2 ri(dom f) regularity.
• Optimality conditions in Fermat form can be established for both the nearly convex problem and its associated convex problem.
• A Lagrange multiplier rule for nearly convex problems follows from the Kuhn-?Tucker conditions of the associated convex problem.
• There exist cases where the nearly convex problem's solution set is not nearly convex, highlighting limitations of the theory.
• The paper provides concrete examples illustrating when the regularity condition fails and the equalities breakdown.