Monotone Objective Function Research Articles

Accurate TDOA estimation using matching cumulative estimator for LFM signal with small samples

This paper introduces two matching cumulative estimators designed for achieving gridless time difference of arrival (TDOA) estimation for linear frequency modulated (LFM) signals with limited samples. A designed matching basis function, customized to the source signal waveform, is employed to calculate with the received data. Time delay matching is then utilized to formulate a monotonic objective function related to the estimated TDOA, enabling gridless estimation. To enhance robustness, especially in scenarios characterized by small samples and low signal to noise ratios (SNRs), a nonlinear cumulative transformation, comprised of a series of strongly convex operations, is incorporated to mitigate the impact of noise on the peak calculation of the objective function. Simulation results validate the efficacy of the two proposed TDOA estimators, particularly under challenging conditions involving small samples and low SNRs.

Targeted Pareto Optimization for Subset Selection With Monotone Objective Function and Cardinality Constraint

Targeted Pareto Optimization for Subset Selection With Monotone Objective Function and Cardinality Constraint

Optimization of partially monotonic functions subject to bipolar fuzzy relation equations

A method to solve a latticed optimization problem constrained by a bipolar fuzzy relation equation is presented in this paper, under the hypothesis of a partially monotonic objective function. The solving strategy consists of transforming the problem into optimizing an order-preserving function in all arguments subject to another bipolar fuzzy relation equation. As a result, all the solutions of the original optimization problem can be deduced from the extremal elements of the feasible domain of the transformed problem. The presented approach embraces the particular case of linear optimization constrained by bipolar fuzzy relation equations.

Stage-t scenario dominance for risk-averse multi-stage stochastic mixed-integer programs

This paper presents a new and general approach, named “Stage-t Scenario Dominance,” to solve the risk-averse multi-stage stochastic mixed-integer programs (M-SMIPs). Given a monotonic objective function, our method derives a partial ordering of scenarios by pairwise comparing the realization of uncertain parameters at each time stage under each scenario. Specifically, we derive bounds and implications from the “Stage-t Scenario Dominance” by using the partial ordering of scenarios and solving a subset of individual scenario sub-problems up to stage t. Using these inferences, we generate new cutting planes to tackle the computational difficulty of risk-averse M-SMIPs. We also derive results on the minimum number of scenario-dominance relations generated. We demonstrate the use of this methodology on a stochastic version of the mean-conditional value-at-risk (CVaR) dynamic knapsack problem. Our computational experiments address those instances that have uncertainty, which correspond to the objective, left-hand side, and right-hand side parameters. Computational results show that our “scenario dominance"-based method can reduce the solution time for mean-risk, stochastic, multi-stage, and multi-dimensional knapsack problems with both integer and continuous variables, whose structure is similar to the mean-risk M-SMIPs, with varying risk characteristics by one-to-two orders of magnitude for varying numbers of random variables in each stage. Computational results also demonstrate that strong dominance cuts perform well for those instances with ten random variables in each stage, and ninety random variables in total. The proposed scenario dominance framework is general and can be applied to a wide range of risk-averse and risk-neutral M-SMIP problems.

An Efficient Evolutionary Algorithm for Subset Selection with General Cost Constraints

In this paper, we study the problem of selecting a subset from a ground set to maximize a monotone objective function f such that a monotone cost function c is bounded by an upper limit. State-of-the-art algorithms include the generalized greedy algorithm and POMC. The former is an efficient fixed time algorithm, but the performance is limited by the greedy nature. The latter is an anytime algorithm that can find better subsets using more time, but without any polynomial-time approximation guarantee. In this paper, we propose a new anytime algorithm EAMC, which employs a simple evolutionary algorithm to optimize a surrogate objective integrating f and c. We prove that EAMC achieves the best known approximation guarantee in polynomial expected running time. Experimental results on the applications of maximum coverage, influence maximization and sensor placement show the excellent performance of EAMC.

Application of the polyblock method to special integer chance constrained problem

The focus in this paper is on a special integer stochastic program with a chance constraint in which, with a given probability, a sum of independent and normally distributed random variables is bounded below. The objective is to maximize the expectation of a linear function of the random variables. The stochastic program is first reduced to an equivalent deterministic integer nonlinear program with monotonic objective and constraints functions. The resulting deterministic is solved using the discrete Polyblock method which exploits its special structure. A numerical example is included for illustration and comparisons with LINGO, COUENNE, BONMIN and BARON solvers are performed.

A revised Variable Neighborhood Search for the Discrete Ordered Median Problem

The paper presents a revised Variable Neighborhood Search (VNS) heuristic method for the Discrete Ordered Median Problem (DOMP). This method introduces a regularization concept that intensifies the searching process for problems with a not strictly monotonic objective function. This allows better quality solutions to be reached, and is especially helpful for the n-center problem. At the same time, the redesigned interchange algorithm is used to boost the computational performance. This serves as the local search and limits the searching process in non-promising directions. It determines new solutions gradually, rejecting those that cannot be better than the current one. In addition, less calculation is required to determine and evaluate new solutions, due to exploiting information from the current solution. Instead of sorting the whole cost vector at each objective function evaluation, the method sorts only the cost components that are actually changing, and updates the ordered cost vector of the current solution. To evaluate the performance, the proposed method is compared with the original VNS for DOMP, along with other existing methods for DOMP from the literature. Exhaustive computational experiments were carried out, utilizing a widely-used set of problem instances from OR-library. The comparison shows that the proposed revised VNS outperforms the other methods, both in computing time and in solution quality.

Finding diverse, high-value representatives on a surface of answers

In many applications, the system needs to selectively present a small subset of answers to users. The set of all possible answers can be seen as an elevation surface over a domain, where the elevation measures the quality of each answer, and the dimensions of the domain correspond to attributes of the answers with which similarity between answers can be measured. This paper considers the problem of finding a diverse set of k high-quality representatives for such a surface. We show that existing methods for diversified top- k and weighted clustering problems are inadequate for this problem. We propose k -DHR as a better formulation for the problem. We show that k -DHR has a submodular and monotone objective function, and we develop efficient algorithms for solving k -DHR with provable guarantees. We conduct extensive experiments to demonstrate the usefulness of the results produced by k -DHR for applications in computational lead-finding and fact-checking, as well as the efficiency and effectiveness of our algorithms.

Production strategy optimization based on iterative discrete Latin hypercube

This paper proposes a new iterative discrete Latin hypercube sampling based method to maximize the objective function (OF) in production strategy optimization. This methodology adequately treats posterior frequency distributions of discrete random variables and maximizes non-necessarily monotonic objective functions within discontinuous search spaces and many local optimums. To validate the method, we used an exhaustive process with an net present value (NPV) proxy, as the objective function, to be maximized. Using as an application case, the benchmark UNISIM-I-D reservoir model, based on Namorado field, Campos basin, Brazil, the method successfully maximized the NPV in the intermediate phase of production strategy optimization, and even compared favorably with a well-established optimization methodology. Population based optimization using discrete Latin hypercube sampling best suited this methodology, with consistent convergence to global optimum, few OF evaluations and the simultaneous multiple numeric reservoir simulations runs. This easy to use, reliable methodology with low computational time costs is an interesting option for optimization methods in problems of production strategy design related to the oil industry.

Monotonic optimization based decoding for linear codes

New efficient methods are developed for the optimal maximum-likelihood (ML) decoding of an arbitrary binary linear code based on data received from any discrete Gaussian channel. The decoding algorithm is based on monotonic optimization that is minimizing a difference of monotonic (d.m.) objective functions subject to the 0---1 constraints of bit variables. The iterative process converges to the global optimal ML solution after finitely many steps. The proposed algorithm's computational complexity depends on input sequence length k which is much less than the codeword length n, especially for a codes with small code rate. The viability of the developed is verified through simulations on different coding schemes.

Dose Finding for Continuous and Ordinal Outcomes with a Monotone Objective Function: A Unified Approach

In many phase I trials, the design goal is to find the dose associated with a certain target toxicity rate. In some trials, the goal can be to find the dose with a certain weighted sum of rates of various toxicity grades. For others, the goal is to find the dose with a certain mean value of a continuous response. In this article, we describe a dose-finding design that can be used in any of the dose-finding trials described above, trials where the target dose is defined as the dose at which a certain monotone function of the dose is a prespecified value. At each step of the proposed design, the normalized difference between the current dose and the target is computed. If that difference is close to zero, the dose is repeated. Otherwise, the dose is increased or decreased, depending on the sign of the difference.

Dose Finding for Continuous and Ordinal Outcomes with a Monotone Objective Function: A Unified Approach

Dose Finding for Continuous and Ordinal Outcomes with a Monotone Objective Function: A Unified Approach

Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.

A gradient based heuristic algorithm and its application to discrete optimization of bar structures

In this paper a nonlinear constrained integer optimization problem with a monotonic objective function is considered. Customarily, the optimum is located near the feasible region boundary for this category of problems. A two-stage heuristic algorithm is developed which utilizes this peculiarity. Within the algorithm coordinate descents are computed to move within the feasible region towards the region boundary. Motions along the boundary are performed using discrete antigradients based on linear approximations of the objective function and constraints at the last feasible point. Auxiliary relative vectors are established to find a better point within a polyhedron formed by hyperplanes tangent to the objective function as well as the violated constraint surfaces. In particular, a model for the optimum design of bar structures is presented. It is demonstrated that both the algorithm and the model have been successfully applied to discrete optimization of ten-bar and two hundred-bar trusses and a single-span two-storey frame.

A relative difference quotient algorithm for discrete optimization

According to the characteristics of discrete optimization, the concept of a relative difference quotient is proposed, and a highly accurate heuristic algorithm, a relative difference quotient algorithm, is developed for a class of discrete optimization problems with monotonic objective functions and constraint functions. The algorithm starts from the minimum point of the objective function outside the feasible region and advances along the direction of minimum increment of the objective function and maximum decrement of constraint functions to find a better approximate optimum solution. In order to evaluate the performance of the algorithm, a stochastic numerical test and a statistical analysis for the test results are also completed. The algorithm has been successfully applied to the discrete optimization of structures.

Approximations to clustering and subgraph problems on trees

Given a tree with vertex weights, we derive fully polynomial approximation schemes for the following two NP-hard problems: (i) Partition the graph into connected cluster sets such that the weight of each set does not exceed a given bound and a monotone objective function is minimized. (ii) Find a subtree with bounded vertex weight such that a monotone objective function is maximized.