This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradientascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is a...

Theminimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one tha...

We consider semidefinite optimization in a saddle point formulation where the primal solution is in the spectrahedron and the dual solution is a distribution over affine functions. We present an approximation algorithm for this problem that runs in sublinear time in the size of the data. To the best of our knowledge, this is the first algorithm to achieve this. Our algorithm is also guaranteed ...

Embedding feature selection in nonlinear SVMs leads to a challenging non-convex minimization problem, which can be prone to suboptimal solutions. This paper develops an effective algorithm to directly solve the embedded feature selection primal problem. We use a trust-region method, which is better suited for non-convex optimization compared to line-search methods, and guarantees convergence to...

An interior path-following algorithm is proposed for solving the nonlinear saddle point problem minimax c T x + (x) + b T y ? (y) ? y T Ax subject to (x; y) 2 X Y R n R m ; where (x) and (y) are smooth convex functions and X and Y are boxes (hyper-rectangles). This problem is closely related to models in stochastic programming and optimal control studied by Rockafellar and Wets. Existence condi...

A small improvement in the structure of the material could save the manufactory a lot of money. The free material design can be formulated as an optimization problem. However, due to its large scale, secondorder methods cannot solve the free material design problem in reasonable size. We formulate the free material optimization (FMO) problem into a saddle-point form in which the inverse of the ...

This paper presents a neural network for solving non-linear minimax multiobjective fractional programming problem subject to nonlinear inequality constraints. Neural model is designed for optimization with constraints condition. Methodology is based on the lagrange multiplier with saddle point optimization.

In this paper, Mond-Weir type duality results for a uncertain multiobjective robust optimization problem are given under generalized invexity assumptions. Also, weak vector saddle-point theorems are obtained under convexity assumptions.

Our main goal in this paper is to show that one can skip gradient computations for gradient descent type methods applied to certain structured convex programming (CP) problems. To this end, we first present an accelerated gradient sliding (AGS) method for minimizing the summation of two smooth convex functions with different Lipschitz constants. We show that the AGS method can skip the gradient...