Additional Assumptions Research Articles

Multilevel Intervention Stepped Wedge Designs (MLI-SWDs).

Multilevel interventions (MLIs) hold promise for reducing health inequities by intervening at multiple types of social determinants of health consistent with the socioecological model of health. In spite of their potential, methodological challenges related to study design compounded by a lack of tools for sample size calculation inhibit their development. We help address this gap by proposing the Multilevel Intervention Stepped Wedge Design (MLI-SWD), a hybrid experimental design which combines cluster-level (CL) randomization using a Stepped Wedge design (SWD) with independent individual-level (IL) randomization. The MLI-SWD is suitable for MLIs where the IL intervention has a low risk of interference between individuals in the same cluster, and it enables estimation of the component IL and CL treatment effects, their interaction, and the combined intervention effect. The MLI-SWD accommodates cross-sectional and cohort designs as well as both incomplete (clusters are not observed in every study period) and complete observation patterns. We adapt recent work using generalized estimating equations for SWD sample size calculation to the multilevel setting and provide an R package for power and sample size calculation. Furthermore, motivated by our experiences with the ongoing NC Works 4 Health study, we consider how to apply the MLI-SWD when individuals join clusters over the course of the study. This situation arises when unemployment MLIs include IL interventions that are delivered while the individual is unemployed. This extension requires carefully considering whether the study interventions will satisfy additional causal assumptions but could permit randomization in new settings.

Unsupervised complex semi-binary matrix factorization for activation sequence recovery of quasi-stationary sources

Industry 5.0 advocates for a sustainable industry, particularly in terms of energy. One of the most fundamental elements of comprehension is knowing when the systems are operated, as this is key to locating energy intensive subsystems and operations. Such knowledge is often lacking in practice. Activation statuses can be recovered from sensor data though. Some non-intrusive sensors (accelerometers, current sensors, etc.) acquire mixed signals containing information about multiple actuators at once. This paper considers different industrial settings in which the identification of binary sub-system activation sequences is sought. In this context, source signals are assumed to be quasi-stationary and they may be correlated. Existing methods either restrict these assumptions, e.g., by imposing orthogonality or noise characteristics, or lift them using additional assumptions, typically using nonlinear transforms.This paper addresses these limitations, and introduces the unsupervised complex semi-binary matrix factorization (ℂSBMF) as its main contribution. In particular, we show that the exact recovery of source activation sequences from non-intrusive sensor data is intrinsically tied to the presence of problematic phase shifts, the causes of which are detailed. A greedy algorithm is proposed, iteratively resynchronizing sources to converge towards the maximum decomposition of each operation despite these phase shifts. The ℂSBMF is verified and compared to existing techniques on synthetic use cases, then validated on experimental data with signals of different nature. To that occasion, the CAFFEINE dataset for unsupervised time series multi-label classification is introduced.

Conformal field theory approach to parton fractional quantum Hall trial wave functions

We show that all lowest Landau-level projected and unprojected chiral parton type fractional quantum Hall ground and edge-state trial wave functions, which take the form of products of integer quantum Hall wave functions, can be expressed as conformal field theory (CFT) correlation functions, where we can associate a chiral algebra to each parton state such that the CFT defined by the algebra is the “smallest” such CFT that can generate the corresponding ground and edge-state trial wave functions (assuming that the corresponding chiral algebra does indeed define a physically “sensible” CFT). A field-theoretic generalization of Laughlin's plasma analogy, known as generalized screening, is formulated for these states. If this holds, along with an additional assumption, we argue that the inner products of edge-state trial wave functions, for parton states where the “densest” trial wave function is unique, can be expressed as matrix elements of an exponentiated local action operator of the CFT, generalizing the result of Dubail [], which implies the equality between edge-state and entanglement level counting to state counting in the corresponding CFT. We numerically test this result in the case of the unprojected ν=2/5 composite fermion state and the bosonic ν=1ϕ22 parton state. We discuss how Read's arguments [] still apply, implying that conformal blocks of the CFT defined by the corresponding chiral algebra are valid quasihole trial wave functions, with the adiabatic braiding statistics given by the monodromy of these functions, assuming the existence of a quasiparticle trapping Hamiltonian. Generalizations of these constructions are discussed, with particular attention given to simple current constructions. It is shown that all chiral composite fermion wave functions can be expressed as CFT correlation functions without explicit symmetrization or antisymmetrization and that the ground, edge, and certain quasihole trial wave functions of the ϕnm parton states can be expressed as the conformal blocks of the U(1)⊗SU(n)m WZW models. Finally, we discuss the relation of the ϕ2k series with the Read-Rezayi series, where several examples of quasihole braiding statistics calculations are given. Published by the American Physical Society 2024

Learning about treatment effects in a new target population under transportability assumptions for relative effect measures.

Investigators often believe that relative effect measures conditional on covariates, such as risk ratios and mean ratios, are "transportable" across populations. Here, we examine the identification of causal effects in a target population using an assumption that conditional relative effect measures are transportable from a trial to the target population. We show that transportability for relative effect measures is largely incompatible with transportability for difference effect measures, unless the treatment has no effect on average or one is willing to make even stronger transportability assumptions that imply the transportability of both relative and difference effect measures. We then describe how marginal (population-averaged) causal estimands in a target population can be identified under the assumption of transportability of relative effect measures, when we are interested in the effectiveness of a new experimental treatment in a target population where the only treatment in use is the control treatment evaluated in the trial. We extend these results to consider cases where the control treatment evaluated in the trial is only one of the treatments in use in the target population, under an additional partial exchangeability assumption in the target population (i.e., an assumption of no unmeasured confounding in the target population with respect to potential outcomes under the control treatment in the trial). We also develop identification results that allow for the covariates needed for transportability of relative effect measures to be only a small subset of the covariates needed to control confounding in the target population. Last, we propose estimators that can be easily implemented in standard statistical software and illustrate their use using data from a comprehensive cohort study of stable ischemic heart disease.

Geometric interpretation of Tensor-Vector-Scalar theory in a Kaluza-Klein reference fluid

Abstract Gravitational alternatives to dark matter require additional fields or assumptions beyond general relativity while continuing to agree with tight solar system constraints. Modified Newtonian Dynamics (MOND), for example, predicts the Tully-Fisher relation for galaxies more accurately than dark matter models while limiting to Newtonian gravity in the solar system. On the other hand, MOND does a poor job predicting larger scale observations such as the Cosmic Microwave Background and Matter Power Spectra. Tensor-Vector-Scalar (TeVeS) theory is a relativistic generalization of MOND that accounts for these observations without dark matter. In this paper, I derive a generalized TeVeS from Kaluza-Klein theory in one extra dimension as a consequence of $n=0$ Kaluza-Klein modes. In the KK theory, MOND is a special case of a slicing condition in the 5D ADM formalism enforced by a reference fluid as in the Isham-Kucha\v{r} method which may arise from a broken displacement symmetry. This has two benefits: first is means that TeVeS is compatible with Kaluza-Klein dark matter theory, which is a strong candidate for Weakly Interacting Massive Particles (WIMPs), the other is that it provides an elegant mechanism for the scalar and vector fields. It constrains most of the freedom in the definition of TeVeS which does not have a field theoretic motivation. This is important because the Kaluza-Klein theory predicts that spin-2 tensor modes must propagate at the speed of light, in agreement with observation, from theoretical constraints while TeVeS has to match this observation empirically. Furthermore, it removes need for the interpolating function in MOND and the Lorentz-violating condition on the vector field to be physical since they are analogous to a gauge condition and depend on state of motion.

Unified framework for the separation property in binary phase-segregation processes with singular entropy densities

Abstract This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$ , without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

The equivalence of Axiom (∗)+ and Axiom (∗)++

Asperó and Schindler have completely solved the Axiom [Formula: see text] vs. [Formula: see text] problem. They have proved that if [Formula: see text] holds then Axiom [Formula: see text] holds, with no additional assumptions. The key question now concerns the relationship between [Formula: see text] and Axiom [Formula: see text]. This is because the foundational issues raised by the problem of Axiom [Formula: see text] vs. [Formula: see text] arguably persist in the problem of Axiom [Formula: see text] vs. [Formula: see text]. The first of our two main theorems is that Axiom [Formula: see text] is equivalent to Axiom [Formula: see text], and as a corollary we show that Axiom [Formula: see text] fails in all the known models of [Formula: see text]. This suggests that [Formula: see text] actually refutes Axiom [Formula: see text]. Our second main theorem is that the [Formula: see text] Conjecture holds assuming [Formula: see text]. This is the strongest partial result known on this conjecture which is one of the central open problems of [Formula: see text]-theory and [Formula: see text]-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both [Formula: see text]-expressible and which implies [Formula: see text], on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the [Formula: see text]-sentences with no parameters. Such sentences are those which simply assert the existence of a set [Formula: see text] for which some property involving only quantification over [Formula: see text] holds.

The Power Word Problem in Graph Products

AbstractThe power word problem for a group $$\varvec{G}$$ G asks whether an expression $$\varvec{u_1^{x_1} \cdots u_n^{x_n}}$$ u 1 x 1 ⋯ u n x n , where the $$\varvec{u_i}$$ u i are words over a finite set of generators of $$\varvec{G}$$ G and the $$\varvec{x_i}$$ x i binary encoded integers, is equal to the identity of $$\varvec{G}$$ G . It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over $$\varvec{G}$$ G ). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group $$\varvec{G}$$ G is $$\varvec{\textsf{uNC}^{1}}$$ uNC 1 -many-one reducible to the power word problem for a finite-index subgroup of $$\varvec{G}$$ G . For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is $$\varvec{\textsf{AC} ^0}$$ AC 0 -Turing-reducible to the word problem for the free group $$\varvec{F_2}$$ F 2 and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups $$\varvec{\mathcal {C}}$$ C without order two elements, the uniform power word problem in a graph product can be solved in $$\varvec{\textsf{AC} ^0[\textsf{C}_=\textsf{L} ^{{{\,\textrm{UPowWP}\,}}(\mathcal {C})}]}$$ AC 0 [ C = L UPowWP ( C ) ] , where $$\varvec{{{\,\textrm{UPowWP}\,}}(\mathcal {C})}$$ UPowWP ( C ) denotes the uniform power word problem for groups from the class $$\varvec{\mathcal {C}}$$ C . As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is $$\varvec{\textsf{NP}}$$ NP -complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.

Maximum principles for elliptic operators in unbounded Riemannian domains

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

The Born rule: Axiom or result?

The Born rule is part of the collapse axioms in the standard version of quantum theory, as presented by most textbooks on the subject. We show here that its signature quadratic dependence on the initial wavefunction's projection onto the measured outcome state follows from a single additional assumption beyond the other axioms. We give two examples of such an assumption, with a separate derivation for each, and we discuss their relationship with existing derivations. Our presentation is suitable for advanced undergraduates or graduate students who have taken a standard course in quantum theory. It does not depend on any particular interpretation of the theory.

Block‐iterative schemes for the split common fixed point problem and applications

In this paper, based on the extrapolated Landweber‐type operators, we present new strongly convergent block‐iterative schemes for solving the split common fixed point problem (SCFPP) with demiclosed strongly quasi‐nonexpansive operators on Hilbert spaces. The strong convergence is proved without the additional assumptions such as the boundedly regular condition and the closedness property of the range of the transformation operator, assumed recently in the literature for the problem. A necessary and sufficient condition that ensures that a th iterate is a solution is given. An application of our results to solve the multiple‐sets split convex feasibility problem (MSSCFP) is showed with computational experiments for illustration.

Analytical models of the electric field and carrier transit time in the base of bipolar junction transistors

The conventional theory of the bipolar junction transistor (BJT) does not clarify the electric field and its effect on the carrier transit in the uniformly doped base region, although the electric field is considered to be a second-order effect at high-level injection. In this study, analytical models of the carrier distribution, electric field, and carrier transit time in the base region are developed without making additional assumptions. It is found that the electric field may change direction, from forward to reverse, as long as the injection level reaches a criterion, affecting the carrier transport behavior. The models are based on the renovated p–n junction theory, provide insight into the carrier transit and are able to cover all levels of injection without the need to handle the low- and high-level injection separately. These models hold significance in retrieving the theoretical integrity of bipolar junction devices, and can be useful for further investigation in structure design, capacitance control, and AC performance optimization of the p–n junction-based device.

An application of the graph approach to life-cycle optimisation of vehicle electrification

Abstract Although durable goods with low energy consumption are being promoted to achieve a decarbonised society, from the perspective of life-cycle assessment, the choice of new durable goods may increase CO2 emissions. To address this problem, research has been conducted on product replacement based on life-cycle optimisation (LCO), a method for identifying a replacement life span that minimises life-cycle CO2 emissions. However, several additional assumptions complicate the analysis of replacement patterns of products and conditional formulas because cumulative emissions do not increase linearly when considering energy mix and technology improvement, and it is difficult to extend the model to optimisation methods in previous LCO studies. This study developed a new LCO approach by applying the shortest path problem to graph theory. Our methodology can contribute to the following: (i) it is computationally inexpensive; (ii) it is intuitively easy to add complex conditions, such as various policy scenarios and parameter changes; and (iii) once the graph of replacement patterns is defined, the optimal solution can be derived using existing solution methods, such as the Dijkstra algorithm. As a case study, we focused on vehicle replacement, which is a major source of CO2 emissions and is being electrified. In particular, we identified vehicle switching paths that minimise life-cycle CO2 emissions by considering changes in Japan’s energy mix and alternative fuel vehicle (AFV) characteristics. We determined that the optimal vehicle replacement path method to reduce CO2 emissions is to switch first to plug-in hybrid electric vehicles (PHEVs) and then to battery electric vehicles (BEVs). Thus, we suggest that the transition to electric vehicles requires a step-by-step process. This methodology is not only conducive to AFV deployment for decarbonisation but can also be applied to other products, such as air conditioners and lighting. Thus, various transition policies could be formulated using our methodology.

Remarks on the existence of minimal models of log canonical generalized pairs

Given an NQC log canonical generalized pair (X,B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,B+M)$$\\end{document} whose underlying variety X is not necessarily Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document}-factorial, we show that one may run a (KX+B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(K_X+B+M)$$\\end{document}-MMP with scaling of an ample divisor which terminates, provided that (X,B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,B+M)$$\\end{document} has a minimal model in a weaker sense or that KX+B+M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_X+B+M$$\\end{document} is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance when the boundary contains an ample divisor.

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

Bilinear optimal control of a one‐dimensional degenerate parabolic equation with a nonlocal term

In this paper, a one‐dimensional degenerate parabolic equation involving a multiplicative control with a nonlocal space term represented by an integral kernel is considered. After some well‐posedness results, we derive several properties of the control‐to‐state mapping. We establish the existence of at least one optimal control and derive the first and the second conditions for optimality, giving the local uniqueness of an optimal control. Finally, we prove that we may expect the global uniqueness of the optimal control under additional assumptions.

Testing the postulates of quantum mechanics with coherent states of light and homodyne detection

Abstract Quantum mechanics has withstood every experimental test thus far. However, it relies on ad-hoc postulates which require experimental verification. Over the past decade there has been a great deal of research testing these postulates, with numerous tests of Born’s rule for determining probabilities and the complex nature of the Hilbert space being carried out. Although these tests are yet to reveal any significant deviation from textbook quantum theory, it remains important to conduct such tests in different configurations and using different quantum states. Here we perform the first such test using coherent states of light in a three-arm interferometer combined with homodyne detection. Our proposed configuration requires additional assumptions, but importantly allows us to use quantum states which exist in a larger Hilbert space compared to previous tests. For testing Born’s rule, we find that the third order interference is bounded to be κ = 0.002 ± 0.004 and for testing whether quantum mechanics is complex or not we find a Peres parameter of F = 1.0000 ± 0.0003 (F = 1 corresponds to the expected complex quantum mechanics). We also design and implement a test of Glauber’s theory of optical coherence.

Breast density quantification in dual-energy mammography using virtual anthropomorphic phantoms.

Breast density is a significant risk factor for breast cancer and can impact the sensitivity of screening mammography. Area-based breast density measurements may not provide an accurate representation of the tissue distribution, therefore volumetric breast density (VBD) measurements are preferred. Dual-energy mammography enables volumetric measurements without additional assumptions about breast shape. In this work we evaluated the performance of a dual-energy decomposition technique for determining VBD by applying it to virtual anthropomorphic phantoms. The dual-energy decomposition formalism was used to quantify VBD on simulated dual-energy images of anthropomorphic virtual phantoms with known tissue distributions. We simulated 150 phantoms with volumes ranging from 50 to 709mL and VBD ranging from 15% to 60%. Using these results, we validated a correction for the presence of skin and assessed the method's intrinsic bias and variability. As a proof of concept, the method was applied to 14 sets of clinical dual-energy images, and the resulting breast densities were compared to magnetic resonance imaging (MRI) measurements. Virtual phantom VBD measurements exhibited a strong correlation (Pearson's ) with nominal values. The proposed skin correction eliminated the variability due to breast size and reduced the bias in VBD to a constant value of -2%. Disagreement between clinical VBD measurements using MRI and dual-energy mammography was under 10%, and the difference in the distributions was statistically non-significant. VBD measurements in both modalities had a moderate correlation (Spearman's =0.68). Our results in virtual phantoms indicate that the material decomposition method can produce accurate VBD measurements if the presence of a third material (skin) is considered. The results from our proof of concept showed agreement between MRI and dual-energy mammography VBD. Assessment of VBD using dual-energy images could provide complementary information in dual-energy mammography and tomosynthesis examinations.

A novel application of data-consistent inversion to overcome spurious inference in genome-wide association studies.

The genome-wide association studies (GWAS) typically use linear or logistic regression models to identify associations between phenotypes (traits) and genotypes (genetic variants) of interest. However, the use of regression with the additive assumption has potential limitations. First, the normality assumption of residuals is the one that is rarely seen in practice, and deviation from normality increases the Type-I error rate. Second, building a model based on such an assumption ignores genetic structures, like, dominant, recessive, and protective-risk cases. Ignoring genetic variants may result in spurious conclusions about the associations between a variant and a trait. We propose an assumption-free model built upon data-consistent inversion (DCI), which is a recently developed measure-theoretic framework utilized for uncertainty quantification. This proposed DCI-derived model builds a nonparametric distribution on model inputs that propagates to the distribution of observed data without the required normality assumption of residuals in the regression model. This characteristic enables the proposed DCI-derived model to cover all genetic variants without emphasizing on additivity of the classic-GWAS model. Simulations and a replication GWAS with data from the COPDGene demonstrate the ability of this model to control the Type-I error rate at least as well as the classic-GWAS (additive linear model) approach while having similar or greater power to discover variants in different genetic modes of transmission.

Reliability-based design optimization for a vertical-type breakwater with multiple limit-state equations under Korean marine environments varying from sea to sea

In this study, as part of basic research aimed at enhancing the accuracy of load and resistance coefficients to ensure their suitability for practical design and promoting the application of underutilized reliability-based design in Korea, the author conducts optimal design based on the reliability analysis of a vertical-type breakwater in the seas off of Haeundae, Yeosu, Mokpo, Gunsan, and Incheon—representative ports in Korea. In doing so, the author utilized the double-loop approach, which simultaneously addresses a reliability problem nested within an optimization process, employing the Polak–He optimization algorithm. To mitigate the substantial numerical effort required by the double-loop approach based on the Polak–He optimization algorithm, which necessitates the gradients of both cost and constraint functions, the subset simulation method was employed. In this process, the author deliberately refrained from using design waves of a specific return period and linear probabilistic models such as the Gaussian distribution, especially concerning wave and lift forces, often viewed as barriers to the widespread application of reliability-based design in Korea. Instead, the author focused on characterizing the uncertainties associated with the wave force, lift force, and overturning moment—variables that significantly impact the integrity of vertical-type breakwaters—by developing probabilistic models for these random variables directly from long-term in situ wave data. These models capture the varied characteristics of the Korean marine environment from sea to sea. In this way, the need for additional assumptions concerning the interrelationship between significant wave and maximum wave heights, along with the wave period, can be eliminated. Following Occam's razor principle, which suggests that explanations constructed with the smallest possible set of assumptions are superior, the reliability-based design optimization of a vertical-type breakwater presented in this study demonstrates promise in terms of simplicity and practicality. The limit state of the vertical-type breakwater was defined to encompass sliding, overturning, and collapse failures, and the strong interrelations between the wave force, lift force, and overturning moment were described using the Nataf joint distribution. As anticipated, simulation results show that solely considering sliding failure, as in the current reliability-based design platform in Korea, leads to an underestimated failure probability. Furthermore, ensuring a consistent failure probability for vertical-type breakwaters using design waves with a specific return period, as in past studies, is not feasible. In contrast, this study demonstrated that breakwaters optimally designed to meet the reliability index requirement of β = 3.5–4 consistently maintain a target failure probability in all sea areas.