On approximate solutions for robust convex semidefinite optimization problems

In this paper, we consider approximate solutions (\(\epsilon \)-solutions) for a convex semidefinite programming problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we prove an approximate optimality theorem and approximate duality theorems for \(\epsilon \)-solutions in robust convex semidefinite programming problem under the robust characteristic cone constraint qualification. Moreover, an example is given to illustrate the obtained results.     

Stability results for properly quasi convex vector optimization problems

In this paper, we discuss the stability of the sets of (weak) minimal points and (weak) efficient points of vector optimization problems. Assuming that the objective functions are (strictly) properly quasi convex, and the data ofthe approximate problems converges to the data of the original problems in the sense of Painlevé–Kuratowski, we establish the Painlevé–Kuratowski set convergence of the sets of (weak) minimal points and (weak) efficient points of the approximate problems to the corresponding ones of original problem. Our main results improve and extend the results of the recent papers.     

On {\epsilon} -solutions for convex optimization problems with uncertainty data

We consider ${\epsilon}$ -solutions (approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish an optimality theorem and duality theorems for ${\epsilon}$ -solutions for the robust convex optimization problem. Moreover, we give an example illustrating the duality theorems.     

Some characterizations of robust optimal solutions for uncertain convex optimization problems

In this paper, we consider robust optimal solutions for a convex optimization problem in the face of data uncertainty both in the objective and constraints. By using the properties of the subdifferential sum formulae, we first introduce a robust-type subdifferential constraint qualification, and then obtain some completely characterizations of the robust optimal solution of this uncertain convex optimization problem. We also investigate Wolfe type robust duality between the uncertain convex optimization problem and its uncertain dual problem by proving duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. Moreover, we show that our results encompass as special cases some optimization problems considered in the recent literature.     

On $$\ell $$-regular bipartitions modulo $$\ell $$

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

     

Weighted robust optimality of convex optimization problems with data uncertainty

In this paper, we introduce the weighted robust counterpart of convex optimization problem with data uncertainty both in the objective and constraints. The     

Limit problems for a Fractional p-La placian as $${ p\to\infty}$$

The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain \({\Omega \subset \mathbb{R}^N}\),

$$\left\{\begin{array}{lll}-\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{\alpha p}}\;dy=f(x,u)\;\;&x\in \Omega,\\ u=g(x) &x\in\mathbb{R}^N\setminus \Omega,\end{array}\right.$$

where \({\alpha\in(0,1)}\) and the exponent p goes to infinity. In particular we will analyze the cases:

1. (i)
   \({f=f(x).}\)
    
2. (ii)
   \({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
    

We show the convergence of the solutions to certain limit as \({p\to\infty}\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:

$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$

where

$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$

     

On one-dimensional superlinear indefinite problems involving the $$\phi $$-Laplacian

Let \(\Omega := ( a,b ) \subset \mathbb {R}\), \(m\in L^{1} ( \Omega ) \) and \(\phi :\mathbb {R\rightarrow R}\) be an odd increasing homeomorphism. We consider the existence of positive solutions for problems of the form

$$\begin{aligned} \left\{ \begin{array} [c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$

where \(f: [ 0,\infty ) \rightarrow [ 0,\infty ) \) is a continuous function which is, roughly speaking, superlinear with respect to \(\phi \). Our approach combines the Guo-Krasnoselski? fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case \(m\ge 0\).

     

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.     

On the approximability of adjustable robust convex optimization under uncertainty

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.     

\epsilon -Henig proper efficiency of set-valued optimization problems in real ordered linear spaces

The aim of this paper is to investigate \(\epsilon \) -Henig proper efficiency of set-valued optimization problems in linear spaces. Firstly, a new notion of \(\epsilon \) -Henig properly efficient point is introduced in linear spaces. Secondly, scalarization theorems of set-valued optimization problems are established in the sense of \(\epsilon \) -Henig proper efficiency. Finally, under the assumption of generalized cone subconvexlikeness, Lagrange multiplier theorems are obtained. Our results generalize some known results in the literature from topological spaces to linear spaces.     

Selected topics in robust convex optimization

Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-but- bounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control.      

On quasi $${\mathcal{H}}$$ -closed subspaces

In this paper some properties concerning the notion of quasi H{\mathcal{H}} -closedness in topological spaces are obtained. Locally quasi H{\mathcal{H}} -closed spaces are studied. Properties of these types of spaces are investigated, when the subsets involved are α-open, semi-open, clopen, or preopen.     

Solving structured nonsmooth convex optimization with complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$

This paper describes an algorithm for solving structured nonsmooth convex optimization problems using the optimal subgradient algorithm (OSGA), which is a first-order method with the complexity \(\mathcal {O}(\varepsilon ^{-2})\) for Lipschitz continuous nonsmooth problems and \(\mathcal {O}(\varepsilon ^{-1/2})\) for smooth problems with Lipschitz continuous gradient. If the nonsmoothness of the problem is manifested in a structured way, we reformulate the problem so that it can be solved efficiently by a new setup of OSGA (called OSGA-V) with the complexity \(\mathcal {O}(\varepsilon ^{-1/2})\). Further, to solve the reformulated problem, we equip OSGA-O with an appropriate prox-function for which the OSGA-O subproblem can be solved either in a closed form or by a simple iterative scheme, which decreases the computational cost of applying the algorithm for large-scale problems. We show that applying the new scheme is feasible for many problems arising in applications. Some numerical results are reported confirming the theoretical foundations.     

On maximal green sequences for type $$\mathbb {}$$ quivers

Given a framed quiver, i.e., one with a frozen vertex associated with each mutable vertex, there is a concept of green mutation, as introduced by Keller. Maximal sequences of such mutations, known as maximal green sequences, are important in representation theory and physics as they have numerous applications, including the computations of spectrums of BPS states, Donaldson–Thomas invariants, tilting of hearts in derived categories, and quantum dilogarithm identities. In this paper, we study such sequences and construct a maximal green sequence for every quiver mutation equivalent to an orientation of a type \(\mathbb {A}\) Dynkin diagram.     

Likelihood robust optimization for data-driven problems

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.     

On the matrix equation $$ X\bar X = \bar XX $$

It is shown that every solution to the equation $ X\bar X = \bar XX $ X\bar X = \bar XX can be reduced by a real orthogonal similarity transformation to a block triangular form with diagonal blocks of orders one and two. If the solution X is a normal matrix, then its block triangular form is actually a block diagonal. In this case, the form of the diagonal blocks is found, yielding new proof of the recent results of Goodson and Horn.     

Stability results for convex vector-valued optimization problems

In this paper, we discuss the stability of the sets of efficient points of vector-valued optimization problems when the data of the approximate problems converges to the data of the original problem in the sense of Painlevé–Kuratowski. Our results improve the corresponding results obtained by Lucchetti and Miglierina (Optimization 53(5–6):517–528, 2004, Section 3).