Many problems in machine learning involve objective functions that are piecewise smooth [7] due to the occurrence of absolute values mins and maxes in their evaluation procedures. See e.g. [8]. For such function we derived in [3] first order (KKT) and second order (SSC) optimality conditions, which can be checked on the basis of a local piecewise linearization [2] that can be computed in an AD ...

In this research, a numerical method by piecewise approximated method for solving fuzzy differential equations is introduced. In this method, the solution by piecewise fuzzy polynomial is present. The base of this method is using fuzzy Taylor expansion on initial value of fuzzy differential equations. The existence, uniqueness and convergence of the approximate solution are investigated. To sho...

In the past few decades, predator–prey model has played an important role in dynamic behavior of populations. Many scholars have studied stability system. Due to complex influence time delay on systems, time-delay systems garnered wide interest. this paper, a classical piecewise smooth slow–fast is considered. The properties system are analyzed by linearization. existence and uniqueness relaxat...

This paper presents a new method for real-time optimization of process systems with a decentralized structure where the idea is to improve computational efficiency and transparency of a solution. The contribution lies in the application and assessment of the Lagrange relaxation and the Dantzig–Wolfe methods, which allows us to efficiently decompose a real-time optimization problem. Furthermore,...

In this paper, an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever for some s > 0, the solution can be approximated to accuracy O(n−s) in energy norm by a continuous piecewise linear function on some partition with n triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate ...

Abstract. We present an analysis for a mixed finite element method for the bending problem of Koiter shell. We derive an error estimate showing that when the geometrical coefficients of the shell mid-surface satisfy certain conditions the finite element method has the optimal order of accuracy, which is uniform with respect to the shell thickness. Generally, the error estimate shows how the acc...

This paper is concerned with the integer quadratic multidimensional knapsack problem (QMKP) where the objective function is separable. Our objective is to determine which expansion technique of the integer variables is the most appropriate to solve (QMKP) to optimality using the upper bound method proposed by Quadri et al. (2007). To the best of our knowledge the upper bound method previously m...