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In this paper, we employ the concept of quasi-relative interior to analyze the method of Lagrange multipliers and establish strong Lagrangian duality for nonsmooth convex optimization problems in Hilbert spaces. Then, we generalize the…
In this paper, the problem of finding optimal success probabilities of static linear optics quantum gates is linked to the theory of convex optimization. It is shown that by exploiting this link, upper bounds for the success probability of…
We examine online safe multi-agent reinforcement learning using constrained Markov games in which agents compete by maximizing their expected total rewards under a constraint on expected total utilities. Our focus is confined to an episodic…
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…
A dual control problem is presented for the optimal stochastic control of a system governed by partial differential equations. Relationships between the optimal values of the original and the dual problems are investigated and two duality…
Semidefinite programs (SDPs) play a crucial role in control theory, traditionally as a computational tool. Beyond computation, the duality theory in convex optimization also provides valuable analytical insights and new proofs of classical…
There are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing…
Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify…
State-of-the-art techniques for simultaneous localization and mapping (SLAM) employ iterative nonlinear optimization methods to compute an estimate for robot poses. While these techniques often work well in practice, they do not provide…
In previous articles, we formalized the problem of optimal allocation strategies for a (perfect) vaccine in an infinite-dimensional metapopulation model. The aim of the current paper is to illustrate this theoretical framework with multiple…
We investigate Lagrangian duality for nonconvex optimization problems. To this aim we use the $\Phi$-convexity theory and minimax theorem for $\Phi$-convex functions. We provide conditions for zero duality gap and strong duality. Among the…
We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…
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 objective of this work is to study the existence of Lagrange multipliers for infinite dimensional problems under G\^ateux differentiability assumptions on the data. Our investigation follows two main steps: the proof of the…
A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost…
In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the…
We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer (1999) for optimal expected utility from nonnegative random variables to the level of optimal…
A unified model is addressed for general optimization problems in multi-scale complex systems. Based on necessary conditions and basic principles in physics, the canonical duality-triality theory is presented in a precise way to include…
Population games can be regarded as a tool to study the strategic interaction of a population of players. Although several attention has been given to such field, most of the available works have focused only on the unconstrained case. That…
We show that the value function of an optimal stopping game driven by a one-dimensional diffusion can be characterised using a modification of the Legendre transformation if and only if the optimal stopping game exhibits a Nash equilibrium…