相关论文: A Counterexample and Fix to a Minimum Distance Dua…
The optimal weak transport problem has recently been introduced by Gozlan et.\ al. We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in…
We develop a synthetic, variational framework for deriving comparison principles in infinite-dimensional Banach spaces. Unlike traditional approaches that rely on the regularity of minimizers and Euler--Lagrange equations, our method…
This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex…
The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double-min duality is solved…
We investigate the continuous non-monotone DR-submodular maximization problem subject to a down-closed convex solvable constraint. Our first contribution is to construct an example to demonstrate that (first-order) stationary points can…
In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational…
The paper studies a general norm minimization problem on a product of normed vector spaces. We establish dual necessary and sufficient optimality conditions and derive explicit formulas for the corresponding solution sets. These formulas…
In this paper we propose a counterexample to the validity of the Comparison Principle and of the Sub and Supersolution Method for nonlocal problems like the stationary Kirchhoff Equation. This counterexample shows that in general smooth…
We establish the Subgradient Theorem for monotone correspondences -- a monotone correspondence is equal to the subdifferential of a potential if and only if it is conservative, i.e. its integral along a closed path vanishes irrespective of…
Usually, bilevel optimization problems need to be transformed into single-level ones in order to derive optimality conditions and solution algorithms. Among the available approaches, the replacement of the lower-level problem by means of…
In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with an asymptotically vanishing damping term. The system under consideration…
For a primal-dual pair of conic linear problems that are described by convex cones $S\subset X$, $T\subset Y$, bilinear symmetric objective functions $\langle\cdot,\cdot\rangle_X$, $\langle\cdot,\cdot\rangle_Y$ and a linear operator…
The notion of duality -- that a given physical system can have two different mathematical descriptions -- is a key idea in modern theoretical physics. Establishing a duality in lattice statistical mechanics models requires the construction…
This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the…
Many results in probability (most famously, Strassen's theorem on stochastic domination), characterize some relationship between probability distributions in terms of the existence of a particular structured coupling between them. Optimal…
We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex…
We prove the existence of global minimizers to the double minimization problem \[ \inf\Big\{ P(E) + \lambda W_p(\mathcal{L}^n \lfloor \, E,\mathcal{L}^n \lfloor\, F) \colon |E \cap F| = 0, \, |E| = |F| = 1\Big\}, \] where $P(E)$ denotes the…
A key idea in convex optimization theory is to use well-structured affine functions to approximate general functions, leading to impactful developments in conjugate functions and convex duality theory. This raises the question: what are the…
By applying the perturbation function approach, we propose the Lagrangian and the conjugate duals for minimization problems of the sum of two, generally nonconvex, functions. The main tools are the $\Phi$-convexity theory and minimax…
We establish a variant of Monge--Kantorovich duality for a constrained optimal transport problem with a continuum of agents, a finite set of alternatives, and general linear constraints. As an application, we revisit the large-market model…