arXiv subjects

Vladimir Shikhman

Publications and source records attributed to Vladimir Shikhman.

At least 19 recordsLinked to original sources

arXiv2026

Solving mathematical programs with complementarity constraints by disjunctive regularizations

We propose a new disjunctive regularization for mathematical programs with complementarity constraints (MPCC). Its feasible set coincides with that of the Kanzow-Schwartz regularization. However, their functional descriptions differ considerably. For the disjunctive regularization, the logical operator OR and equivalent max-type constraints are used. Unlike the Kanzow-Schwartz, the disjunctive regularization satisfies the tailored linear independence constraint qualification if the original MPCC does. More than that, the favorable convergence properties - known to hold for the Kanzow-Schwartz regularization - remain valid for the disjunctive regularization as well. In particular, no second order necessary conditions are required to guarantee convergence towards S-stationary points of MPCC. Additionally, we keep track of the topological type of approximating and limiting nondegenerate C-stationary points in terms of their C-indices. Quadratic and biactive parts of the C-indices are shown to generically correspond to each other while regularizing. This is a new phenomenon as compared to the Scholtes or sign-type regularizations studied before. Numerical experiments illustrate that the proposed disjunctive regularization clearly outperforms the Kanzow-Schwartz regularization. Its numerical performance is even better than that of the Scholtes regularization if solving MPCCs with high accuracy.

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arXiv2025

Ramsey model of optimal growth with Allee effect

For the Ramsey model of economic growth, which describes the optimal allocation of consumption and saving over time, we assume the population dynamics to follow the Allee effect. The so-called Allee threshold separates two regimes from each other. If starting below the threshold, the population decreases to zero. Above this threshold, it gradually saturates. We show that the corresponding consumption per capita stabilizes at two different levels accordingly. As for our main result, the consumption per capita performs in the long run better if the population becomes extinct, rather then it advances the saturation level. This is in line with our previous results on the capital per capita for the underlying Solow-Swan model of economic growth with the Allee effect. However, the comparison of consumption-to-capital ratios at the both steady states crucially depends on the curvature of the production function.

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arXiv2025

On stability of the Scholtes regularization for mathematical programs with complementarity constraints

For mathematical programs with complementarity constraints (MPCC), we study the stability properties of their Scholtes regularization. Our goal is to relate nondegenerate C-stationary points of MPCC with nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization up to their topological type. As it is standard in the framework of Morse theory, the topological types are captured by the C-index and the quadratic index, respectively. It turns out that a change of the topological type for the approximating Karush-Kuhn-Tucker points of the Scholtes regularization and their limiting C-stationary point is possible. In particular, a minimizer of MPCC with zero C-index might be approximated by saddle points of the Scholtes regularization with nonzero quadratic index. In order to bypass this index shift phenomenon, an additional generic condition for nondegenerate C-stationary points of MPCC is identified. It says that non-biactive multipliers under consideration should not vanish. Then, we uniquely trace nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization and successively maintain the topological type of their limiting C-stationary point. The main technical issue here is to relate the first-order information of the defining functions, which enters the biactive part of the C-index, with the second-order information, which enters the quadratic index of the Karush-Kuhn-Tucker points.

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arXiv2024

Stationarity in nonsmooth optimization between geometrical motivation and topological relevance

The goal of this paper is to compare alternative stationarity notions in structured nonsmooth optimization (SNO). Here, nonsmoothness is caused by complementarity, vanishing, orthogonality type, switching, or disjunctive constraints. On one side, we consider geometrically motivated notions of $\widehat N$-, $N$-, and $\overline{N}$-stationarity in terms of Fréchet, Mordukhovich, and Clarke normal cones to the feasible set, respectively. On the other side, we advocate the notion of topologically relevant T-stationarity, which adequately captures the global structure of SNO. Our main findings say that (a) $\widehat N$-stationary points include all local minimizers; (b) $N$-stationary points, which are not $\widehat N$-stationary, correspond to the singular saddle points of first order; (c) T-stationary points, which are not $N$-stationary, correspond to the regular saddle points of first order; (d) $\overline{N}$-stationary points, which are not T-stationary, are irrelevant for optimization purposes. Overall, a hierarchy of stationarity notions for SNO is established.

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arXiv2024

Mathematics of Family Planning in Talmud

Motivated by the commitments from the Talmud in Judaism, we consider the family planning rules which require a couple to get children till certain numbers of boys and girls are reached. For example, the rabbinical school of Beit Hillel says that one boy and one girl are necessary, whereas Beit Shammai urges for two boys. Surprisingly enough, although the corresponding average family sizes differ in both cases, the gender ratios remain constant. We show more that for any family planning rule the gender ratio is equal to the birth odds. The proof of this result is given by using different mathematical techniques, such as induction principle, Doob's optional-stopping theorem, and brute-force. We conclude that, despite possible asymmetries in the religiously motivated family planning rules, they discriminate neither boys nor girls.

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arXiv2024

Scalarization via utility functions in multi-objective optimization

We study a general scalarization approach via utility functions in multi-objective optimization. It consists of maximizing utility which is obtained from the objectives' bargaining with regard to a disagreement reference point. The theoretical framework for a broad class of utility functions from microeconomics is developed. For that, we associate a utility-dependent single-objective optimization problem with the given multi-objective optimization problem. We show that Pareto optimal points of the latter can be recovered by solving the former. In particular, Cobb-Douglas, Leontief, and CES utility functions are considered. We prove that any Pareto optimal point can be obtained as a solution of scalarization via one of the mentioned utility functions. Further, we propose a numerical scheme to solve utility-dependent single-objective optimization problems. Here, the main difficulty comes from the necessity to address constraints which are associated with a disagreement reference point. Our crucial observation is that the explicit treatment of these additional constraints may be avoided. This is the case if the Slater condition is satisfied and the utility function under consideration has the so-called barrier property. Under these assumptions, we prove the convergence of our scheme to Pareto optimal points. Numerical experiments on real-world financial datasets in a portfolio selection context confirm the efficiency of our scalarization approach via utility functions.

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arXiv2022

Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

We extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an intermediate step the recently introduced regularized continuous reformulation of a cardinality-constrained optimization problem. We show that the Scholtes-type regularization method is well-defined locally around a nondegenerate T-stationary point of this regularized continuous reformulation. Moreover, the nondegenerate Karush-Kuhn-Tucker points of the corresponding Scholtes-type regularization converge to a T-stationary point having the same index, i.e. its topological type persists. Overall, we conclude that the global structure of the regularized continuous reformulation and its Scholtes-type regularization essentially coincide.

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arXiv2022

Global aspects of the continuous reformulation for cardinality-constrained optimization problems

The main goal of this paper is to relate the topologically relevant stationary points of a cardinality-constrained optimization problem and its continuous reformulation up to their type. For that, we focus on the nondegenerate M- and T-stationary points, respectively. Their so-called M- and T-indices, which uniquely determine the global and local structure of optimization problems under consideration in algebraic terms, are traced. As novelty, we suggest to regularize the continuous reformulation for this purpose. The main consequence of our analysis is that the number of saddle points of the regularized continuous reformulation grows exponentially as compared to that of the initial cardinality-constrained optimization problem. Additionally, we obtain the Morse theory for the regularized continuous reformulation by using the corresponding results on mathematical programs with orthogonality type constraints.

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arXiv2022

On local uniqueness of normalized Nash equilibria

For generalized Nash equilibrium problems (GNEP) with shared constraints we focus on the notion of normalized Nash equilibrium in the nonconvex setting. The property of nondegeneracy for normalized Nash equilibria is introduced. Nondegeneracy refers to GNEP-tailored versions of linear independence constraint qualification, strict complementarity and second-order regularity. Surprisingly enough, nondegeneracy of normalized Nash equilibrium does not prevent from degeneracies at the individual players' level. We show that generically all normalized Nash equilibria are nondegenerate. Moreover, nondegeneracy turns out to be a sufficient condition for the local uniqueness of normalized Nash equilibria. We emphasize that even in the convex setting the proposed notion of nondegeneracy differs from the sufficient condition for (global) uniqueness of normalized Nash equilibria, which is known from the literature.

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arXiv2021

Optimality conditions for mathematical programs with orthogonality type constraints

We consider the class of mathematical programs with orthogonality type constraints (MPOC). Orthogonality type constraints appear by reformulating the sparsity constraint via auxiliary binary variables and relaxing them afterwards. For MPOC a necessary optimality condition in terms of T-stationarity is stated. The justification of T-stationarity is threefold. First, it allows to capture the global structure of MPOC in terms of Morse theory, i. e. deformation and cell-attachment results are established. For that, nondegeneracy for the T-stationary points is introduced and shown to hold at a generic MPOC. Second, we prove that Karush-Kuhn-Tucker points of the Scholtes-type regularization converge to T-stationary points of MPOC. This is done under the MPOC-tailored linear independence constraint qualification (LICQ), which turns out to be a generic property too. Third, we show that T-stationarity applied to the relaxation of sparsity constrained nonlinear optimization (SCNO) naturally leads to its M-stationary points. Moreover, we argue that all T-stationary points of this relaxation become degenerate.

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arXiv2021

Network manipulation algorithm based on inexact alternating minimization

In this paper, we present a network manipulation algorithm based on an alternating minimization scheme from (Nesterov 2020). In our context, the latter mimics the natural behavior of agents and organizations operating on a network. By selecting starting distributions, the organizations determine the short-term dynamics of the network. While choosing an organization in accordance with their manipulation goals, agents are prone to errors. This rational inattentive behavior leads to discrete choice probabilities. We extend the analysis of our algorithm to the inexact case, where the corresponding subproblems can only be solved with numerical inaccuracies. The parameters reflecting the imperfect behavior of agents and the credibility of organizations, as well as the condition number of the network transition matrix have a significant impact on the convergence of our algorithm. Namely, they turn out not only to improve the rate of convergence, but also to reduce the accumulated errors. From the mathematical perspective, this is due to the induced strong convexity of an appropriate potential function.

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arXiv2021

Regularized quasi-monotone method for stochastic optimization

We adapt the quasi-monotone method from [2] for composite convex minimization in the stochastic setting. For the proposed numerical scheme we derive the optimal convergence rate in terms of the last iterate, rather than on average as it is standard for subgradient methods. The theoretical guarantee for individual convergence of the regularized quasi-monotone method is confirmed by numerical experiments on l1-regularized robust linear regression.

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arXiv2021

Cardinality-constrained optimization problems in general position and beyond

We study cardinality-constrained optimization problems (CCOP) in general position, i. e. those optimization-related properties that are fulfilled for a dense and open subset of their defining functions. We show that the well-known cardinality-constrained linear independence constraint qualification (CC-LICQ) is generic in this sense. For M-stationary points we define nondegeneracy and show that it is a generic property too. In particular, the sparsity constraint turns out to be active at all minimizers of a generic CCOP. Moreover, we describe the global structure of CCOP in the sense of Morse theory, emphasizing the strength of the generic approach. Here, we prove that multiple cells need to be attached, each of dimension coinciding with the proposed M-index of nondegenerate M-stationary points. Beyond this generic viewpoint, we study singularities of CCOP. For that, the relation between nondegeneracy and strong stability in the sense of Kojima (1980) is examined. We show that nondegeneracy implies the latter, while the reverse implication is in general not true. To fill the gap, we fully characterize the strong stability of M-stationary points under CC-LICQ by first- and second-order information of CCOP defining functions. Finally, we compare nondegeneracy and strong stability of M-stationary points with second-order sufficient conditions recently introduced in the literature.

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arXiv2021

Dynamic pricing under nested logit demand

Recently, there is growing interest and need for dynamic pricing algorithms, especially, in the field of online marketplaces by offering smart pricing options for big online stores. We present an approach to adjust prices based on the observed online market data. The key idea is to characterize optimal prices as minimizers of a total expected revenue function, which turns out to be convex. We assume that consumers face information processing costs, hence, follow a discrete choice demand model, and suppliers are equipped with quantity adjustment costs. We prove the strong smoothness of the total expected revenue function by deriving the strong convexity modulus of its dual. Our gradient-based pricing schemes outbalance supply and demand at the convergence rates of $\mathcal{O}(\frac{1}{t})$ and $\mathcal{O}(\frac{1}{t^2})$, respectively. This suggests that the imperfect behavior of consumers and suppliers helps to stabilize the market.

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arXiv2020

Topological approach to mathematicalprograms with switching constraints

We study mathematical programs with switching constraints (MPSC)from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous defor-mation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian's Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy andstrong stability of W-stationary points. A complete characterizationof strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, all bi-active Lagrange multipliers of a strongly stable W-stationary point cannot vanish.

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arXiv2020

On nondegenerate M-stationary points for sparsity constrained nonlinear optimization

We study sparsity constrained nonlinear optimization (SCNO) from a topological point of view. Special focus will be on M-stationary points from Burdakov et al. (2016). We introduce nondegenerate M-stationary points and define their M-index. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic SCNO. Some relations to other stationarity concepts, such as S-stationarity, basic feasibility, and CW-minimality, are discussed in detail. By doing so, the issues of instability and degeneracy of points due to different stationarity concepts are highlighted. The concept of M-stationarity allows to adequately describe the global structure of SCNO along the lines of Morse theory. For that, we study topological changes of lower level sets while passing an M-stationary point. As novelty for SCNO, multiple cells of dimension equal to the M-index are needed to be attached. This intriguing fact is in strong contrast with other optimization problems considered before, where just one cell suffices. As a consequence, we derive a Morse relation for SCNO, which relates the numbers of local minimizers and M-stationary points of M-index equal to one. The appearance of such saddle points cannot be thus neglected from the perspective of global optimization. Due to the multiplicity phenomenon in cell-attachment, a saddle point may lead to more than two different local minimizers. We conclude that the relatively involved structure of saddle points is the source of well-known difficulty if solving SCNO to global optimality.

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arXiv2020

Critical point theory for sparse recovery

We study the problem of sparse recovery in the context of compressed sensing. This is to minimize the sensing error of linear measurements by sparse vectors with at most $s$ non-zero entries. We develop the so-called critical point theory for sparse recovery. This is done by introducing nondegenerate M-stationary points which adequately describe the global structure of this nonconvex optimization problem. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic sparse recovery problem. Additionally, the equivalence of strong stability and nondegeneracy for M-stationary points is shown. We claim that the appearance of saddle points - these are M-stationary points with exactly $s-1$ non-zero entries - cannot be neglected. For this purpose we derive a so-called Morse relation, which gives a lower bound on the number of saddle points in terms of the number of local minimizers. The relatively involved structure of saddle points can be seen as a source of well-known difficulty by solving the problem of sparse recovery to global optimality.

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arXiv2019

Discrete choice prox-functions on the simplex

We derive new prox-functions on the simplex from additive random utility models of discrete choice. They are convex conjugates of the corresponding surplus functions. In particular, we explicitly derive the convexity parameter of discrete choice prox-functions associated with generalized extreme value models, and specifically with generalized nested logit models. Incorporated into subgradient schemes, discrete choice prox-functions lead to natural probabilistic interpretations of the iteration steps. As illustration we discuss an economic application of discrete choice prox-functions in consumer theory. The dual averaging scheme from convex programming naturally adjusts demand within a consumption cycle.

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