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We investigate the convergence of the primal-dual algorithm for composite optimization problems when the objective functions are weakly convex. We introduce a modified duality gap function, which is a lower bound of the standard duality gap function. Under the sharpness condition of this new function, we identify the area around the set of saddle points where we obtain the convergence of the primal-dual algorithm. We give numerical examples and applications in image denoising and deblurring to demonstrate our results.

We prove sufficient and necessary conditions ensuring zero duality gap for Lagrangian duality in some classes of nonconvex optimization problems. To this aim, we use the $\Phi$-convexity theory and minimax theorems for $\Phi$-convex functions. The obtained zero duality results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance., Comment: arXiv admin note: substantial text overlap with arXiv:2011.09194

The paper concerns clustering with respect to the shape and size of 2D contours that are boundaries of cross-sections of 3D objects of revolution. We propose a number of similarity measures based on combined disparate Procrustes analysis (PA) and Dynamic Time Warping (DTW) distances. Motivation and the main application for this study comes from archaeology. The performed computational experiments refer to the clustering of archaeological pottery.

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 theorems for $\Phi$-convex functions. We provide conditions ensuring zero duality gap and introduce $\Phi$-Karush-Kuhn-Tucker conditions that characterize solutions to primal and dual problems. We also discuss the relationship between the dual problems introduced in the present investigation and some conjugate-type duals existing in the literature.

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 classes of functions, to which our duality results can be applied, are prox-bounded functions, DC functions, weakly convex functions and paraconvex functions.

We use the framework of a type of abstract convexity ($\Phi_{lsc}$-convexity) to investigate properties of lower semicontinuous quadratically minorized functions in Hilbert spaces. A new result, which states that, for every local $\Phi_{lsc}$-subgradient there exists a global one is proved and plays a crucial role in our considerations. We deliver conditions for abstract subdifferentiability ($\Phi_{lsc}$-subdifferentiability) of locally $C^{1,1}$ functions, twice continuously differentiable functions, prox-regular functions and paraconvex functions. As an application we establish a new sufficient and necessary condition for minimax equality for $\Phi_{lsc}$-convex functions. This new condition is expressed in therms of $\Phi_{lsc}$-subdifferential.

In this paper provide sufficient and necessary conditions for the minimax equality for extended-valued $\Phi$-convex functions. As an application we establish sufficient and necessary conditions for the minimax equality for convex-concave functions., Comment: This paper has been withdrawn by the author due to major changes in article and title

We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\Phi$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\Phi$-convex functions. These conditions are expressed in terms of abstract $\Phi$-subgradients., Comment: 13 pages