A general duality theory for nonsmooth cone-constrained optimization

Many real-world optimization problems in engineering, economics, and control systems involve nonsmooth objectives and complex constraints that violate classical regularity assumptions. Standard duality theories often fail in these settings, particularly when dealing with nonconvexity, nondifferentiability, or complementarity-type structures. This paper develops a generalized duality framework for cone-constrained optimization problems based on two mild assumptions: calmness of the constraint mapping and -pseudoinvexity of the objective function. These conditions enable strong, weak, and converse duality theorems without requiring classical constraint qualifications. Using Mordukhovich subdifferential calculus, we derive optimality conditions and construct dual models that avoid active-set identification and product-type constraints. While mathematical programs with vanishing constraints (MPVC) are recovered as special cases, our framework is substantially more general. Beyond the theoretical contributions, we design a primal–dual subgradient algorithm grounded in nonsmooth variational principles. We prove its convergence and demonstrate its practical effectiveness through illustrative numerical examples.