Related papers: Optimality conditions for minimizers at infinity i…
This paper is devoted to study of optimality conditions at infinity in nonsmooth minimax programming problems and applications. By means of the limiting subdifferential and normal cone at infinity, we dirive necessary and sufficient…
We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some representations of polynomials non-negative on a basic semi-algebraic set, defined by polynomial inequalities, under more general conditions. Based on each…
Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let…
Consider a polynomial optimization problem. Adding polynomial equations generated by the Fritz John conditions to the constraint set does not change the optimal value. As proved in [arXiv:2205.04254 (2022)], the objective polynomial has…
This paper is devoted to establishing an enhanced Fritz John type first-order necessary condition for a general constrained nonlinear infinite-dimensional optimization problem. Unlike traditional constraint qualifications in optimization…
We consider the semi-infinite optimization problem: $f^*:=\min_{x\in X}\:\{f(x): g(x,y)\,\leq \,0,\:\forally\in Y_x\}$, where $f,g$ are polynomials and $X\subset R^n$ as well as $Y_\x\subset R^p$, $x\in X$, are compact basic semi-algebraic…
The main objective of this paper is to derive the optimality conditions for one type of fuzzy optimization problems. At the beginning, we define a cone of descent direction for fuzzy optimization, and prove that its intersection with the…
Let $f,g_1,\dots,g_m$ be polynomials of degree at most $d$ with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Assume that $f$ is non-negative on a basic semi-algebraic set $S$ defined by polynomial inequalities $g_j(x)\ge…
In this paper we summarize our results in infinite horizon optimal control. We present optimality conditions for weak local minimizer in the framework of weighted functions. Moreover we formulate the Pontryagin Maximum Principle for strong…
We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our…
We consider the inverse optimization problem associated with the polynomial program f^*=\min \{f(x): x\in K\}$ and a given current feasible solution $y\in K$. We provide a systematic numerical scheme to compute an inverse optimal solution.…
This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem…
Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point…
The paper extends the widely used in optimisation theory decoupling techniques to infinite collections of functions. Extended concepts of uniform lower semicontinuity and firm uniform lower semicontinuity are discussed. The main theorems…
Let $F \in \R[X_1,\ldots,X_n]$ and the zero set $V=\zero(\mathcal{P},\R^n)$, where $\mathcal{P}:=\{P_1,\ldots,P_s\} \subset \R[X_1,\ldots,X_n]$ is a finite set of polynomials. We investigate existence of critical points of $F$ on an…
In this paper, we establish the existence of the efficient solutions for polynomial vector optimization problems on a nonempty closed constraint set without any convexity and compactness assumptions. We first introduce the relative…
We consider an optimization problem in a convex space $E$ with an affine objective function, subject to $J$ constraints in the forms of inequalities on some other affine functions, where $J$ is a given nonnegative integer. Under suitable…
We develop refined Karush-Kuhn-Tucker (KKT) and Fritz-John (FJ)-type optimality conditions for nonsmooth, nonconvex mathematical pro\-gra\-mming problems. We pay special attention in the case that the functional constraint belongs to a…
The purpose of this paper is to characterize the weak efficient solutions, the efficient solutions, and the isolated efficient solutions of a given vector optimization problem with finitely many convex objective functions and infinitely…
Study about theory and algorithms for constrained optimization usually assumes that the feasible region of the optimization problem is nonempty. However, there are many important practical optimization problems whose feasible regions are…