凸积分泛函的广义极小元，Bregman距离，平方关系式
Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities
摘要：
基于凸法向被积函数的积分泛函被最小化，受到许多矩约束。被积函数在正数上是有限的，在负数上是无限的，是严格凸的但不一定是可微的。在凸对偶的框架下，将最小化问题作为原问题与对偶问题一起研究。值函数的有效域用锥核来描述，锥核是对早期凸核概念的修正。极小化和广义极小化是显式地构造从修改的对偶问题的解决方案，不假设原始约束规格。利用Bregman距离和被积函数缺乏本质光滑性的修正项，给出了一个推广的平方关系式.结果被应用到Bregman距离的最小化。在对偶约束规格的条件下，只要对偶值是有限的，则广义对偶解的存在性成立。“不正常”情况的例子也包括在内，指出某些关键结果的普遍性的局限性。
英文摘要：
Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers and generalized minimizers are explicitly constructed from solutions of modified dual problems, not assuming the primal constraint qualification. A generalized Pythagorean identity is presented using Bregman distance and a correction term for lack of essential smoothness in integrands. Results are applied to minimization of Bregman distances. Existence of a generalized dual solution is established whenever the dual value is finite, assuming the dual constraint qualification. Examples of `irregular' situations are included, pointing to the limitations of generality of certain key results.
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