arXiv subjects
Publications and source records attributed to Tien-Son Pham.
In this paper we present necessary and sufficient conditions (in terms of Łojasiewicz inequalities) for the stability of local minimum points in smooth unconstrained optimization. In particular, we derive a sufficient condition for which the local minimum property of a given function is determined by its Taylor polynomial of a certain degree.
This paper identifies necessary and sufficient conditions for the exactness of penalty functions in optimization problems whose constraint sets are not necessarily bounded. The case where the data of problems is locally Lipschitz, semi-algebraic or non-degenerate polynomials is studied in detail. The conditions are given in terms of properties of the objective and residual functions of the problems in question. The obtained results generalize and improve some known results in the literature on exact penalty functions.
This paper, in the setting at infinity, presents some relationships between the modulus of metric regularity and the radius of (strong) metric regularity that gives a measure of the extent to which a set-valued mapping can be perturbed before (strong) metric regularity is lost. The results given here can be viewed as versions at infinity of [2, Theorem 1.5] and [3, Theorem 4.6].
We first study Clarke's tangent cones at infinity to unbounded subsets of $\mathbb{R}^n.$ We prove that these cones are closed convex and show a characterization of their interiors. We then study subgradients at infinity for extended real value functions on $\mathbb{R}^n$ and derive necessary optimality conditions at infinity for optimization problems. We also give a number of rules for the computing of subgradients at infinity and provide some characterizations of the Lipschitz continuity at infinity for lower semi-continuous functions.
We present a global version of the Łojasiewicz inequality on comparing the rate of growth of two polynomial functions in the case the mapping defined by these functions is (Newton) non-degenerate at infinity. In addition, we show that the condition of non-degeneracy at infinity is generic in the sense that it holds in an open and dense semi-algebraic set of the entire space of input data.
Let $f$ be a real polynomial function with $n$ variables and $S$ be a basic closed semialgebraic set in $\Bbb{R}^n$. In this paper, we are interested in the problem of identifying the type (local minimizer, maximizer or not extremum point) of a given isolated KKT point $x^*$ of $f$ over $S.$ To this end, we investigate some properties of the tangency variety of $f$ on $S$ at $x^*,$ by which we introduce the definition of faithful radius of $f$ over $S$ at $x^*.$ Then, we show that the type of $x^*$ can be determined by the global extrema of $f$ over the intersection of $S$ and the Euclidean ball centered at $x^*$ with a faithful radius. Finally, we propose an algorithm involving algebraic computations to compute a faithful radius of $x^*$ and determine its type.
Given a semialgebraic set-valued map $F \colon \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ with closed graph, we show that the map $F$ is Holder metrically subregular and that the following conditions are equivalent: (i) $F$ is an open map from its domain into its range and the range of $F$ is locally closed; (ii) the map $F$ is Holder metrically regular; (iii) the inverse map $F^{-1}$ is pseudo-Holder continuous; (iv) the inverse map $F^{-1}$ is lower pseudo-Holder continuous. An application, via Robinson's normal map formulation, leads to the following result in the context of semialgebraic variational inequalities: if the solution map (as a map of the parameter vector) is lower semicontinuous then the solution map is finite and pseudo-Hölder continuous. In particular, we obtain a negative answer to a question mentioned in the paper of Dontchev and Rockafellar \cite{Dontchev1996}. As a byproduct, we show that for a (not necessarily semialgebraic) continuous single-valued map from $\mathbb{R}^n$ to $\mathbb{R},$ the openness and the non-extremality are equivalent. This fact improves the main result of Pühn \cite{Puhl1998}, which requires the convexity of the map in question.
This paper is devoted to present new error bounds of regularized gap functions for polynomial variational inequalities with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. The developed techniques are largely based on variational analysis and semialgebraic geometry, which allow us to establish a nonsmooth extension of the seminal Łojasiewicz's gradient inequality to regularized gap functions with explicitly calculated exponents.
Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so--called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and sufficient conditions for $\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\bar{x}$ is an isolated local minimizer of $f,$ we define a "tangency exponent" $α_* > 0$ so that for any $α\in \mathbb{R}$ the following four conditions are always equivalent: (i) the inequality $α\ge α_*$ holds; (ii) the point $\bar{x}$ is an $α$th order sharp local minimizer of $f;$ (iii) the limiting subdifferential $\partial f$ of $f$ is $(α- 1)$th order strongly metrically subregular at $\bar{x}$ for $0;$ and (iv) the function $f$ satisfies the Łojaseiwcz gradient inequality at $\bar{x}$ with the exponent $1 - \frac{1}α.$ Besides, we also present a counterexample to a conjecture posed by Drusvyatskiy and Ioffe [Math. Program. Ser. A, 153(2):635--653, 2015].
Given polynomial maps $f, g \colon \mathbb{R}^n \to \mathbb{R}^n,$ we consider the {\em polynomial complementary problem} of finding a vector $x \in \mathbb{R}^n$ such that \begin{equation*} f(x) \ \ge \ 0, \quad g(x) \ \ge \ 0, \quad \textrm{ and } \quad \langle f(x), g(x) \rangle \ = \ 0. \end{equation*} In this paper, we present various properties on the solution set of the problem, including genericity, nonemptiness, compactness, uniqueness as well as error bounds with exponents explicitly determined. These strengthen and generalize some previously known results, and hence broaden the boundary knowledge of nonlinear complementarity problems as well.
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a hierarchy of semidefinite programming (SDP) relaxations and prove the convergence result for the method. Numerical experiments are presented to show the efficiency of the proposed algorithm.
Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\em tangency variety} of $f$ on $S$: (i) The $f$ is bounded from below on $S;$ (ii) The $f$ attains its infimum on $S;$ (iii) The sublevel set $\{x \in S \ | \ f(x) \le λ\}$ for $λ\in \mathbb{R}$ is compact; (iv) The $f$ is coercive on $S.$ Besides, we also provide some stability criteria for boundedness and coercivity of $f$ on $S.$
We construct an invariant of the bi-Lipschitz contact equivalence of continuous function germs definable in a polynomially bounded o-minimal structure, such as semialgebraic functions. For a germ $f,$ the invariant is given in terms of the leading coefficients of the asymptotic expansions of $f$ along the connected components of the tangency variety of $f.$
The paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint sets. The main attention is paid to the two major notions of optimality in vector problems: Pareto efficiency and proper efficiency in the sense of Geoffrion. Employing adequate tools of variational analysis and generalized differentiation, we first establish relationships between the notions of properness, $M$-tameness, and the Palais--Smale conditions formulated for the restriction of the vector cost mapping on the constraint set. These results are instrumental to derive verifiable necessary and sufficient conditions for the existence of Pareto efficient solutions in vector optimization. Furthermore, the developed approach allows us to obtain new sufficient conditions for the existence of Geoffrion-properly efficient solutions to such constrained vector problems.
We are interested in the existence of Pareto solutions to the vector optimization problem $$\text{Min}_{\,\mathbb{R}^m_+} \{f(x) \,|\, x\in \mathbb{R}^n\},$$ where $f\colon\mathbb{R}^n\to \mathbb{R}^m$ is a polynomial map. By using the {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m - 1$ containing the set of Pareto values of the problem. Then we establish connections between the Palais--Smale conditions, $M$-tameness, and properness for the map $f$. Based on these results, we provide some sufficient conditions for the existence of Pareto solutions of the problem. We also introduce a generic class of polynomial vector optimization problems having at least one Pareto solution.
Let $S\subset \mathbb{C}^n$ be a non-singular algebraic set and $f \colon \mathbb{C}^n \to \mathbb{C}$ be a polynomial function. It is well-known that the restriction $f|_S \colon S \to \mathbb{C}$ of $f$ on $S$ is a locally trivial fibration outside a finite set $B(f|_S) \subset \mathbb{C}.$ In this paper, we give an explicit description of a finite set $T_\infty(f|_S) \subset \mathbb{C}$ such that $B(f|_S) \subset K_0(f|_S) \cup T_\infty(f|_S),$ where $K_0(f|_S)$ denotes the set of critical values of the $f|_S.$ Furthermore, $T_\infty(f|_S)$ is contained in the set of critical values of certain polynomial functions provided that the $f|_S$ is Newton non-degenerate at infinity. Using these facts, we show that if $\{f_t\}_{t \in [0, 1]}$ is a family of polynomials such that the Newton polyhedron at infinity of $f_t$ is independent of $t$ and the $f_t|_S$ is Newton non-degenerate at infinity, then the global monodromies of the $f_t|_S$ are all isomorphic.
In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are {\em not necessarily reduced}. We first show that the polar quotients is a topological invariant. We next prove that the Łojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. As an application, we give effective estimates of the Łojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.
In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point $c$, which is degenerate, of a multivariate polynomial function $f$. To this end, we introduce the definition of faithful radius of $c$ by means of the curve of tangency of $f$. We show that the type of $c$ can be determined by the global extrema of $f$ over the Euclidean ball centered at $c$ with a faithful radius.We propose algorithms to compute a faithful radius of $c$ and determine its type.